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How to Solve for Both X & Y

How to Solve a System of Equations

Solving for two variables (normally denoted as "x" and "y") requires two sets of equations. Assuming you have two equations, the best way for solving for both variables is to use the substitution method, which involves solving for one variable as far as possible, then plugging it back in to the other equation. Knowing how to solve a system of equations with two variables is important for several areas, including trying to find the coordinate for points on a graph.

Write out the two equations that have the two variables you want to solve. For this example, we will find the value for "x" and "y" in the two equations "3x + y = 2" and "x + 5y = 20"

Solve for one of the variables in on one of the equations. For this example, let's solve for "y" in the first equation. Subtract 3x from each side to get "y = 2 - 3x"

Plug in the y value found from the first equation in to the second equation in order to find the x value. In the previous example, this means the second equation becomes "x + 5(2- 3x) = 20"

Solve for x . The example equation becomes "x + 10 - 15x = 20," which is then "-14 x + 10 = 20." Subtract 10 from each side, divide by 14 and you have end up with x = -10/14, which simplifies to x = -5/7.

Plug in the x value in to the first equation to find out the y value. y = 2 - 3(-5/7) becomes 2 + 15/7, which is 29/7.

Check your work by plugging in the x and y values in to both of the equations.

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Drew Lichtenstein started writing in 2008. His articles have appeared in the collegiate newspaper "The Red and Black." He holds a Master of Arts in comparative literature from the University of Georgia.

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System of Equations

In mathematics, a system of equations, also known as a set of simultaneous equations or an equation system, is a finite set of equations for which we sought common solutions. In systems of equations, variables are related in a specific way in each equation. i.e., the equations can be solved simultaneously to find a set of values of variables that satisfies each equation.

System of linear equations finds applications in our day-to-day lives in modelling problems where the unknown values can be represented in form of variables. Solving system of equations involves different methods such as substitution, elimination, graphing, etc. Let us look into each method in detail.

What is a System of Equations?

In algebra, a system of equations comprises two or more equations and seeks common solutions to the equations. "A system of linear equations is a set of equations which are satisfied by the same set of values of variables."

System of Equations Example

A system of equations as discussed above is a set of equations that seek a common solution for the variables included. The following set of linear equations is an example of the system of equations:

• 2x - y = 12
• x - 2y = 48

Note that the values x = -8 and y = -28 satisfy each of these equations and hence the pair (x, y) = (-8, -28) is the solution of the above system of equations. But how to solve system of equations? Let us see.

Solutions of System of Equations

Solution of system of equations is the set of values of variables that satisfies each linear equation in the system. The main reason behind solving an equation system is to find the value of the variable that satisfies the condition of all the given equations true. There systems of equations are classified into 3 types depending on their number of solutions:

• Linear System with "Unique solution"
• Linear System with "No solution"
• Linear System with "Infinitely many solutions"

We know that every linear equation represents a line on the coordinate plane. In this perception, the above figure should give more sense to understanding the different types of solutions of system of equations.

Unique Solution of a System of Equations

A system of equations has unique solution when there is only set of variables exist that satisfy each equation in the system. In terms of graphs, a system with a unique solution has lines (representing the equations) that intersect (at one point).

No Solution

A system of equations has no solution when there exists no set of variables that satisfy each linear equation in the system. If we graph that kind of system, the resulting lines will be parallel to each other.

Infinite Many Solutions

A system of equations can have infinitely many solutions when there exist infinite sets of variables that satisfy each equation. In such cases, the lines corresponding to the linear equations would overlap each other on the graph. i.e., both equations represent the same line. Since a line has infinite points on it, each point on it becomes a solution of the system.

Solving System of Equations

Solving a system of equations means finding the values of the variables used in the set of equations. Any system of equations can be solved in different methods.

• Substitution Method
• Elimination Method
• Graphical Method
• Cross-multiplication method

To solve a system of equations in 2 variables, we need at least 2 equations. Similarly, for solving a system of equations in 3 variables, we will require at least 3 equations. Let us understand 3 ways to solve a system of equations given the equations are linear equations in two variables.

☛ Also Check: Solving System of Linear Equations

Solving System of Linear Equations By Substitution Method

For solving the system of equations using the substitution method given two linear equations in x and y, in one of the equations, express y in terms x in one of the equations and then substitute it in the other equation.

Example: Solve the system of equations using the substitution method.

3x − y = 23 → (1)

4x + 3y = 48 → (2)

From (1), we get:

y = 3x − 23 → 3

Plug in y in (2),

4x + 3 (3x − 23) = 48

13x − 69 = 48

⇒x = 9

Now, plug in x = 9 in (1)

y = 3 × 9 − 23 = 4

Hence, x = 9 and y = 4 is the solution of the given system of equations.

Solving System of Equations By Elimination Method

Using the elimination method to solve the system of equations, we eliminate one of the unknowns, by multiplying equations by suitable numbers, so that the coefficients of one of the variables become the same.

Example: Solve the following system of linear equations by elimination method.

2x + 3y = 4 → (1) and 3x + 2y = 11 → (2)

The coefficients of y are 3 and 2; LCM (3, 2) = 6

Multiplying Equation (1) by 2 and Equation (2) by 3, we get

4x + 6y = 8 → (3)

9x + 6y = 33 → (4)

On subtracting (3) from (4), we get

⇒x = 5

Plugging in x = 5 in (2) we get

15 + 2y = 11

⇒y = −2

Hence, x = 5, y = −2 is the solution.

Solving System of Equations by Graphing

In this method, solving system of linear equations is done by plotting their graphs. "The point of intersection of the two lines is the solution of the system of equations using graphical method ."

Example: 3x + 4y = 11 and -x + 2y = 3

Find at least two values of x and y satisfying equation 3x + 4y = 11

So we have 2 points A (1, 2) and B (3, 0.5).

Similarly, find at least two values of x and y satisfying the equation -x + 2y = 3

We have two points C(-3, 0) and D( 3, 3). Plotting these points on the graph we can get the lines in a coordinate plane as shown below.

We observe that the two lines intersect at (1, 2). So, x = 1, y = 2 is the solution of given system of equations. Methods I and II are the algebraic way of solving simultaneous equations and III is the graphical method .

Solving System of Equations Using Cross-multiplication method

In this method, we solve the systems of equations a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0 using the cross-multiplication formula:

x / (b 1 c 2 - b 2 c 1 ) = -y / (a 1 c 2 - a 2 c 1 ) = 1 / (a 1 b 2 - a 2 b 1 )

For more detailed information of this method, click here .

Solving System of Equations Using Matrices

The solution of a system of equations can be solved using matrices . In order to solve a system of equations using matrices, express the given equations in standard form , with the variables and constants on respective sides. for the given equations,

a\(_1\)x + \(b_1\)y + \(c_1\)z = \(d_1\)

a\(_2\)x + \(b_2\)y + \(c_2\)z = \(d_2\)

a\(_3\)x + \(b_3\)y + \(c_3\)z = \(d_3\)

we can express them in the form of matrices as,

\(\left[\begin{array}{ccc} a_1x + b_1y + c_1 z \\ a_2x + b_2y + c_2 z \\ a_3x + b_3y + c_3 z \end{array}\right] = \left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

⇒\(\left[\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right] + \left[\begin{array}{ccc} x \\ y \\ z \end{array}\right] = \left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

⇒ AX = B

A = \(\left[\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array}\right]\), X = \(\left[\begin{array}{ccc} x \\ y \\ z \end{array}\right]\), B = \(\left[\begin{array}{ccc} d_1 \\ d_2 \\ d_3 \end{array}\right]\)

⇒ The solution of the system is given by the formula X = A -1 B, where A -1 = inverse of the matrix A. To understand this method more in detail, click here .

Alternatively, Cramer's rule can also be used to solve the system of equations using determinants .

Applications of System of Equations

Systems of equations are a very useful tool and find application in our day-to-day lives for modeling real-life situations and analyzing questions about them.

• System of Equations Word Problems
• Systems of Equations Worksheet

For applying the concept of the system of equations, we need to translate the given situation into two linear equations in two variables , then further solve to find the solution of linear programming problems. Any method to solve the system of equations, substitution, elimination, graphical, etc methods. Follow the below-given steps to apply the system of equations to solve problems in our daily lives,

• To translate and represent the given situation in form of a system of equations, identify unknown quantities in a problem and represent them with variables.
• Write a system of equations modelling the conditions of the problem.
• Solve the system of equations.
• Check and express the obtained solution in terms of the given context.

☛Related Articles:

Check these articles related to the concept of the system of equations.

• Solutions of a Linear Equation
• Simultaneous Linear Equations
• Solving Linear Equations Calculator
• Equation Calculator
• System of Equations Calculator

System of Equations Examples

Example 1: In a ΔLMN, ∠N = 3∠M = 2(∠L+∠M). Calculate the angles of ΔLMN using this information.

Let ∠L = x° and ∠M = y°.

∠N = 3∠M = (3y)°.

∠L + ∠M + ∠N = 180° ∴ x + y + 3y = 180 x + 4y = 180 ... (1) From the given equation, 3∠M = 2(∠L+∠M) ∴ 3y = 2(x + y) ⇒ 2x - y = 0 ... (2) On multiplying (2) by 4 and adding the result to (1), we get 9x = 180 x = 20 Substitute this in (2): 40 - y = 0 ⇒ y = 40 Therefore, ∠L = 20° and ∠M = 40°. Therefore, the third angle is, ∠N = 180 - (20 + 40) = 120°. Answer: ∠L = 20°, ∠M = 40°, and ∠N = 120°.

Example 2: Peter is three times as old as his son. 5 years later, he shall be two and half times as old as his son. What's Peter's present age?

Let Peter's age be x years and his son's age be y years. Then by given info,

x = 3y ... (1)

5 years later,

Peter's age = x + 5 years and his son's age = y + 5 years.

By the given condition,

x + 5 = 5/2 (y + 5)

2x - 5y - 15 = 0 ... (2)

From (1), we have x = 3y. Substitute this in (2):

2(3y) - 5y - 15 = 0

Substitute this in (1): x = 3(15) = 45

Thus, the solution of the given system is (x, y) = (45, 15).

Answer: Present age of Peter = 45 years; Present age of son = 15 years.

Example 3: Tressa starts her job with a certain monthly salary and earns a fixed increment every year. If her salary was $1500 after 4 years and $1800 after 10 years of service, find her starting salary and annual increment.

Let her starting salary be x and annual increment be y.

After 4 years, her salary was $1500

x + 4y = 1500 ... (1)

After 10 years, her salary was $1800

x + 10y = 1800 ... (2)

Subtracting Eqn (1) from (2), we get

∴ y = 50

On putting y = 50 in (1), we get x = 1300.

Answer: Starting salary was $1300 and annual increment is $50.

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Practice Questions on System of Equations

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FAQs on System of Equations

What is a system of equations in mathematics.

A system of equations in mathematics is a set of linear equations that need to be solved to find a common solution. A real-life problem with two or more unknowns can be converted into a system of equations and can be solved to find a set of values of variables that satisfy all equations.

How Do You Solve a System of Equations?

Solving a system of equations is computing the unknown variables still balancing the equations on both sides. We solve an equation system to find the values of the variables that satisfy the condition of all the given equations. There are different methods to solve a system of equations,

• Graphical method
• Substitution method
• Elimination method

How Do You Create a System of Equations with Two Variables?

To create a system of equations with two variables:

• First, identify the two unknown quantities in the given problem.
• Next, find out the two conditions given and frame equations for each of them.

How to Solve a System of Equations by Substitution Method?

The substitution method is one of the ways to solve a system of equations in two variables, given the set of linear equations. In this method, we substitute the value of a variable found by one equation in the second equation.

How to Solve a System of Equations Using the Elimination Method?

The elimination method is used to solve a system of linear equations. In the elimination method, we eliminate one of the two variables by multiplying each equation with the required numbers and try to solve equations with another variable. In this process, finding LCM of coefficients would be helpful.

What is the Purpose of Graphing Systems of Equations?

To solve the system of equations, given a set of linear equations graphically, we need to find at least two solutions for the respective equations. We observe the pattern of lines after plotting the points to infer it is consistent, dependent, or inconsistent.

• If the two lines are intersecting at the same point, then the intersection point gives a unique solution for the system of equations.
• If the two lines coincide, then in this case there are infinitely many solutions.
• If the two lines are parallel, then in this case there is no solution.

What are Homogeneous System of Linear Equations?

The homogeneous system of linear equations is a set of linear equation each one of which has its constant term to be 0. The process of solving these kind of systems can be learnt in detailed by clicking here .

How to Solve a System of Equations With Cross Multiplication Method?

While solving a system of equations using the cross-multiplication method , we use the formula x / (b 1 c 2 - b 2 c 1 ) = -y / (a 1 c 2 - a 2 c 1 ) = 1 / (a 1 b 2 - a 2 b 1 ) to solve the system a 1 x + b 1 y + c 1 = 0 and a 2 x + b 2 y + c 2 = 0.

How do you Solve a System of Equations Using 2 Equations with 3 Variables?

An equation with 3 variables represents a plane .

• Step 1) To solve a system of 2 equations with 3 variables say x, y, and z, we will consider the 1st two equations and eliminate one of the variables, say x, to obtain a new equation.
• Step 2) Next, we write the 2nd variable, y in terms of z from the new equation and substitute it in the third equation.
• Step 3) Assuming z = a, we will obtain values of x and y also in terms of 'a'.
• Step 4) Once, we know the value of a, we can find the values of x and y in terms of 'a'.

What is the Solution to the System of Equations and How to Find It?

Solving system of equations is finding the values of variables that satisfy each equation in that system. There are three main methods to solving system of equations, they are:

Where Can I Find System of Linear Equations Calculator?

The system of linear equations calculator is available here . This allows us to enter the linear equations. Then it will show the solution along with step-by-step solution.

Solving Equations

What is an equation.

An equation says that two things are equal. It will have an equals sign "=" like this:

That equations says:

what is on the left (x − 2)  equals  what is on the right (4)

So an equation is like a statement " this equals that "

What is a Solution?

A Solution is a value we can put in place of a variable (such as x ) that makes the equation true .

Example: x − 2 = 4

When we put 6 in place of x we get:

which is true

So x = 6 is a solution.

How about other values for x ?

• For x=5 we get "5−2=4" which is not true , so x=5 is not a solution .
• For x=9 we get "9−2=4" which is not true , so x=9 is not a solution .

In this case x = 6 is the only solution.

You might like to practice solving some animated equations .

More Than One Solution

There can be more than one solution.

Example: (x−3)(x−2) = 0

When x is 3 we get:

(3−3)(3−2) = 0 × 1 = 0

And when x is 2 we get:

(2−3)(2−2) = (−1) × 0 = 0

which is also true

So the solutions are:

x = 3 , or x = 2

When we gather all solutions together it is called a Solution Set

The above solution set is: {2, 3}

Solutions Everywhere!

Some equations are true for all allowed values and are then called Identities

Example: sin(−θ) = −sin(θ) is one of the Trigonometric Identities

Let's try θ = 30°:

sin(−30°) = −0.5 and

−sin(30°) = −0.5

So it is true for θ = 30°

Let's try θ = 90°:

sin(−90°) = −1 and

−sin(90°) = −1

So it is also true for θ = 90°

Is it true for all values of θ ? Try some values for yourself!

How to Solve an Equation

There is no "one perfect way" to solve all equations.

A Useful Goal

But we often get success when our goal is to end up with:

x = something

In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.

Example: Solve 3x−6 = 9

Now we have x = something ,

and a short calculation reveals that x = 5

Like a Puzzle

In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do.

Here are some things we can do:

• Add or Subtract the same value from both sides
• Clear out any fractions by Multiplying every term by the bottom parts
• Divide every term by the same nonzero value
• Combine Like Terms
• Expanding (the opposite of factoring) may also help
• Recognizing a pattern, such as the difference of squares
• Sometimes we can apply a function to both sides (e.g. square both sides)

Example: Solve √(x/2) = 3

And the more "tricks" and techniques you learn the better you will get.

Special Equations

There are special ways of solving some types of equations. Learn how to ...

• solve Quadratic Equations
• solve Radical Equations
• solve Equations with Sine, Cosine and Tangent

Check Your Solutions

You should always check that your "solution" really is a solution.

How To Check

Take the solution(s) and put them in the original equation to see if they really work.

Example: solve for x:

2x x − 3 + 3 = 6 x − 3     (x≠3)

We have said x≠3 to avoid a division by zero.

Let's multiply through by (x − 3) :

2x + 3(x−3) = 6

Bring the 6 to the left:

2x + 3(x−3) − 6 = 0

Expand and solve:

2x + 3x − 9 − 6 = 0

5x − 15 = 0

5(x − 3) = 0

Which can be solved by having x=3

Let us check x=3 using the original question:

2 × 3 3 − 3 + 3  =   6 3 − 3

Hang On: 3 − 3 = 0 That means dividing by Zero!

And anyway, we said at the top that x≠3 , so ...

x = 3 does not actually work, and so:

There is No Solution!

That was interesting ... we thought we had found a solution, but when we looked back at the question we found it wasn't allowed!

This gives us a moral lesson:

"Solving" only gives us possible solutions, they need to be checked!

• Note down where an expression is not defined (due to a division by zero, the square root of a negative number, or some other reason)
• Show all the steps , so it can be checked later (by you or someone else)

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Wolfram|Alpha is capable of solving a wide variety of systems of equations. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Additionally, it can solve systems involving inequalities and more general constraints.

Learn more about:

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Tips for entering queries

Enter your queries using plain English. To avoid ambiguous queries, make sure to use parentheses where necessary. Here are some examples illustrating how to ask about solving systems of equations.

• solve y = 2x, y = x + 10
• solve system of equations {y = 2x, y = x + 10, 2x = 5y}
• y = x^2 - 2, y = 2 - x^2
• solve 4x - 3y + z = -10, 2x + y + 3z = 0, -x + 2y - 5z = 17
• solve system {x + 2y - z = 4, 2x + y + z = -2, z + 2y + z = 2}
• solve 4 = x^2 + y^2, 4 = (x - 2)^2 + (y - 2)^2
• x^2 + y^2 = 4, y = x
• View more examples

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What are systems of equations?

A system of equations is a set of one or more equations involving a number of variables..

The solutions to systems of equations are the variable mappings such that all component equations are satisfied—in other words, the locations at which all of these equations intersect. To solve a system is to find all such common solutions or points of intersection.

Systems of linear equations are a common and applicable subset of systems of equations. In the case of two variables, these systems can be thought of as lines drawn in two-dimensional space. If all lines converge to a common point, the system is said to be consistent and has a solution at this point of intersection. The system is said to be inconsistent otherwise, having no solutions. Systems of linear equations involving more than two variables work similarly, having either one solution, no solutions or infinite solutions (the latter in the case that all component equations are equivalent).

More general systems involving nonlinear functions are possible as well. These possess more complicated solution sets involving one, zero, infinite or any number of solutions, but work similarly to linear systems in that their solutions are the points satisfying all equations involved. Going further, more general systems of constraints are possible, such as ones that involve inequalities or have requirements that certain variables be integers.

Solving systems of equations is a very general and important idea, and one that is fundamental in many areas of mathematics, engineering and science.

• Solve equations and inequalities
• Simplify expressions
• Factor polynomials
• Graph equations and inequalities
• Advanced solvers
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What can QuickMath do?

QuickMath will automatically answer the most common problems in algebra, equations and calculus faced by high-school and college students.

• The algebra section allows you to expand, factor or simplify virtually any expression you choose. It also has commands for splitting fractions into partial fractions, combining several fractions into one and cancelling common factors within a fraction.
• The equations section lets you solve an equation or system of equations. You can usually find the exact answer or, if necessary, a numerical answer to almost any accuracy you require.
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• \frac{3}{4}x+\frac{5}{6}=5x-\frac{125}{3}
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