How to Create Two-Variable Equations & Inequalities

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will provide instruction on creating equations in two or more variables in order represent relationships between quantities, then graph the equations on coordinate axes with labels and scales.

Writing Two-Variable Equations

Have you ever tried to work out a problem where there were two unknown values? For instance, let's say that you have $5 and plan to save $5 each week. Your friend has $11, but is planning to save $2 each week. After how many weeks will you and your friend have the same amount of money? How much money will you both have? We can use a system of equations to solve this problem.

A system of equations is two or more equations with two variables that can be solved by graphing on the coordinate plane. In order to graph a system of equations, both equations can be changed into the slope-intercept form of a line.


slope intercept


The y-intercept is where you begin on the y-axis and the slope is how much you move by. When graphing the slope, we make it a fraction and then we go up or down (rise) and then we go right (run).

Example 1

To answer the question of when you and your friend will have the same amount of money and how much money you will have, we must first define our variables. Since I know we are going to graph these on the coordinate plane, I will use x and y, but any letter would suffice.

In order to choose which variable is the x and which is the y, we need to determine which one depends on the other. In the problem I am looking for the amount of money and the number of weeks. Does the number of weeks depend on how much money you have or does the amount of money you have depend on the number of weeks?

Since the amount of money you have, depends on the number of weeks, the amount of money is the dependent variable, y and the number of weeks is the independent variable, x.

Now that we have defined our variables, we are going to create two equations. One equation will represent your money, and the second equation will represent your friend's money. The total amount of money you have, y is equal to how much you currently have, $5, plus how much you are saving each week, $5 per week. That translates to y=5x+5.

The total amount of money your friend has, y, is equal to how much he currently has, $11, plus how much he is saving each week, $2 per week. That translates to y=2x+11.

In order to solve this system of equations, we are going to graph both equations on the same coordinate plane. When we graph linear equations, we need a starting point and the slope. The starting point is the amount of money you have, $5, which is also the y-intercept in the equation. The slope is the amount of money you are saving, $5 per week. If we make the slope a fraction, it is 5/1, which means we go up 5 and right 1 from the starting point on the y-axis.


graph 1


Next, we graph the second equation. The starting amount here is $11 and the slope is $2 per week. We start at 11 on the y-axis and then go up 2 and over 1.


graph


To find the solution to this system of equations, we find the point where the two lines intersect. They intersect at the point (2, 15). This means that x=2 and y=15. So after 2 weeks, both you and your friend will have $15.

Example 2

Susan and Carlos went to a carnival, where they had to pay for admission and each ride ticket. Susan paid for four ride tickets and admission, and it cost her $9. Carlos paid for admission and two ride tickets, and it cost him $7. How much did admission cost? How much did each ride ticket cost?

Since we are trying to find the cost of admission and the cost of a ride ticket, we need to define our variables. It doesn't matter in this case which one we use as x and y because these two events are independent of each other. Let's let the cost of admission be our y and the cost of each ride ticket be x.

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