solve a system of equations by graphing word problems calculator

System of Equations: Graphing Method Calculator

Instructions: Use this calculator to solve a system of two linear equations using the graphical method. Please type two valid linear equations in the boxes provided below:

More about the graphing method to solve linear systems

Systems of linear equations are very commonly found in different context of Algebra. The most commonly found systems in basic Algebra courses are 2 by 2 systems, which consist of two lines equations and two variables.

Such two-by-two systems often appear when solving word problems, proportion problems and assignment problems with constraint. Naturally, larger systems (with more variables and equations) also are common, here focus only on 2x2 systems, because those we can graph.

How to use graphing method

The graphing method consists of representing each of the linear equations as a line on a graph. Then, we need to find the intersection points between two lines , using the observation that the intersection point of the line (if it exists) will the solution of the system.

What happens if the intersection does not exist? That would be case if the lines are parallel without being the same line, in which case, there is no intersection. The rule is clear: when there is no intersection between the lines, there is no solution to the system.

There is a third case that can also happen: The lines could be parallel but actually identical (this is, they are the same line). So, how many intersection points do you have? Yes, your guess right: you have infinite intersection points, which means that you have infinite solutions.

Solving Systems of equations by graphing answers

So, the methodology is simple: You start with a linear system, and the first thing you do is to graph the two linear equations .

Then, you look at the graph and assess whether the lines intersect at one point only (which happens if the lines have different slopes, in which case you have a unique solution.

If not, see if they parallel and different, in which case there are no solutions. Otherwise, if the two lines are equal, then we have infinite solutions.

How do you solve a system of equations on a graphing calculator?

All systems have different ways of working. In this case of this graphing calculator, all you have to do is to type two linear equations, even if they are not completely simplified. The calculator first will try to get the lines into slope-intercept and will provide you with a graph and with an approximated estimate of the solution.

Different calculators will provide different outputs, but the great advantage of this calculator is that it will provide all the steps of the process.

How do you write systems of equations from a graph?

Linear functions are univocally connected. This is, one linear equation is associated with one and one line only, and conversely, a line is associated with one linear equation and one linear equation only.

So, in order to write systems of equations from a graph, you need to work with each line separately. Take one line and identify two points on the line. With those two points you can compute the slope of the line .

Then, with the slope of the line and the y-intercept, you can write the equation of the line in slope-intercept form .

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Course: Algebra 2   >   Unit 10

• Solving equations by graphing
• Solving equations by graphing: intro
• Solving equations graphically: intro

Solving equations by graphing: graphing calculator

• Solving equations graphically: graphing calculator
• Solving equations by graphing: word problems
• Solving equations graphically: word problems
• Equations: FAQ

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Video transcript

Graphing Calculator

What do you want to calculate.

• Solve for Variable
• Practice Mode
• Step-By-Step

Example (Click to try)

How to graph your problem.

• Type in your equation like y=2x+1 (If you have a second equation use a semicolon like y=2x+1 ; y=x+3)
• Press Calculate it to graph!

Graphing Equations Video Lessons

• Khan Academy Video: Graphing Lines
• Khan Academy Video: Graphing a Quadratic Function

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Solving Systems of Equations Real World Problems

Wow! You have learned many different strategies for solving systems of equations! First we started with Graphing Systems of Equations . Then we moved onto solving systems using the Substitution Method . In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations.

Now we are ready to apply these strategies to solve real world problems! Are you ready? First let's look at some guidelines for solving real world problems and then we'll look at a few examples.

Steps For Solving Real World Problems

• Highlight the important information in the problem that will help write two equations.
• Define your variables
• Write two equations
• Use one of the methods for solving systems of equations to solve.
• Check your answers by substituting your ordered pair into the original equations.
• Answer the questions in the real world problems. Always write your answer in complete sentences!

Ok... let's look at a few examples. Follow along with me. (Having a calculator will make it easier for you to follow along.)

Example 1: Systems Word Problems

You are running a concession stand at a basketball game. You are selling hot dogs and sodas. Each hot dog costs $1.50 and each soda costs $0.50. At the end of the night you made a total of $78.50. You sold a total of 87 hot dogs and sodas combined. You must report the number of hot dogs sold and the number of sodas sold. How many hot dogs were sold and how many sodas were sold?

1.  Let's start by identifying the important information:

• hot dogs cost $1.50
• Sodas cost $0.50
• Made a total of $78.50
• Sold 87 hot dogs and sodas combined

2.  Define your variables.

• Ask yourself, "What am I trying to solve for? What don't I know?

In this problem, I don't know how many hot dogs or sodas were sold. So this is what each variable will stand for. (Usually the question at the end will give you this information).

Let x = the number of hot dogs sold

Let y = the number of sodas sold

3. Write two equations.

One equation will be related to the price and one equation will be related to the quantity (or number) of hot dogs and sodas sold.

1.50x + 0.50y = 78.50    (Equation related to cost)

 x + y = 87   (Equation related to the number sold)

4.  Solve! 

We can choose any method that we like to solve the system of equations. I am going to choose the substitution method since I can easily solve the 2nd equation for y.

5. Think about what this solution means.

x is the number of hot dogs and x = 35. That means that 35 hot dogs were sold.

y is the number of sodas and y = 52. That means that 52 sodas were sold.

6.  Write your answer in a complete sentence.

35 hot dogs were sold and 52 sodas were sold.

7.  Check your work by substituting.

1.50x + 0.50y = 78.50

1.50(35) + 0.50(52) = 78.50

52.50 + 26 = 78.50

35 + 52 = 87

Since both equations check properly, we know that our answers are correct!

That wasn't too bad, was it? The hardest part is writing the equations. From there you already know the strategies for solving. Think carefully about what's happening in the problem when trying to write the two equations.

Example 2: Another Word Problem

You and a friend go to Tacos Galore for lunch. You order three soft tacos and three burritos and your total bill is $11.25. Your friend's bill is $10.00 for four soft tacos and two burritos. How much do soft tacos cost? How much do burritos cost?

• 3 soft tacos + 3 burritos cost $11.25
• 4 soft tacos + 2 burritos cost $10.00

In this problem, I don't know the price of the soft tacos or the price of the burritos.

Let x = the price of 1 soft taco

Let y = the price of 1 burrito

One equation will be related your lunch and one equation will be related to your friend's lunch.

3x + 3y = 11.25  (Equation representing your lunch)

4x + 2y = 10   (Equation representing your friend's lunch)

We can choose any method that we like to solve the system of equations. I am going to choose the combinations method.

5. Think about what the solution means in context of the problem.

x = the price of 1 soft taco and x = 1.25.

That means that 1 soft tacos costs $1.25.

y = the price of 1 burrito and y = 2.5.

That means that 1 burrito costs $2.50.

Yes, I know that word problems can be intimidating, but this is the whole reason why we are learning these skills. You must be able to apply your knowledge!

If you have difficulty with real world problems, you can find more examples and practice problems in the Algebra Class E-course.

Take a look at the questions that other students have submitted:

Problem about the WNBA

Systems problem about ages

Problem about milk consumption in the U.S.

Vans and Buses? How many rode in each?

Telephone Plans problem

Systems problem about hats and scarves

Apples and guavas please!

How much did Alice spend on shoes?

All about stamps

Going to the movies

Small pitchers and large pitchers - how much will they hold?

Chickens and dogs in the farm yard

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• Systems Word Problems

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About solving equations

A value is said to be a root of a polynomial if ..

The largest exponent of appearing in is called the degree of . If has degree , then it is well known that there are roots, once one takes into account multiplicity. To understand what is meant by multiplicity, take, for example, . This polynomial is considered to have two roots, both equal to 3.

One learns about the "factor theorem," typically in a second course on algebra, as a way to find all roots that are rational numbers. One also learns how to find roots of all quadratic polynomials, using square roots (arising from the discriminant) when necessary. There are more advanced formulas for expressing roots of cubic and quartic polynomials, and also a number of numeric methods for approximating roots of arbitrary polynomials. These use methods from complex analysis as well as sophisticated numerical algorithms, and indeed, this is an area of ongoing research and development.

Systems of linear equations are often solved using Gaussian elimination or related methods. This too is typically encountered in secondary or college math curricula. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Similar remarks hold for working with systems of inequalities: the linear case can be handled using methods covered in linear algebra courses, whereas higher-degree polynomial systems typically require more sophisticated computational tools.

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• \frac{3}{4}x+\frac{5}{6}=5x-\frac{125}{3}
• \sqrt{2}x-\sqrt{3}=\sqrt{5}
• 7y+5-3y+1=2y+2
• \frac{x}{3}+\frac{x}{2}=10
• What is a linear equation?
• A linear equation represents a straight line on a coordinate plane. It can be written in the form: y = mx + b where m is the slope of the line and b is the y-intercept.
• How do you find the linear equation?
• To find the linear equation you need to know the slope and the y-intercept of the line. To find the slope use the formula m = (y2 - y1) / (x2 - x1) where (x1, y1) and (x2, y2) are two points on the line. The y-intercept is the point at which x=0.
• What are the 4 methods of solving linear equations?
• There are four common methods to solve a system of linear equations: Graphing, Substitution, Elimination and Matrix.
• How do you identify a linear equation?
• Here are a few ways to identify a linear equation: Look at the degree of the equation, a linear equation is a first-degree equation. Check if the equation has two variables. Graph the equation.
• What is the most basic linear equation?
• The most basic linear equation is a first-degree equation with one variable, usually written in the form of y = mx + b, where m is the slope of the line and b is the y-intercept.

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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.

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• 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
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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.