How To Solve Systems Of Equations By Graphing

Systems of equations can help solve real-life questions in all kinds of fields, from chemistry to business to sports. Solving them isn't just important for your math grades; it can save you a lot of time whether you're trying to set goals for your business or your sports team.

For example, imagine you and your friend are setting up a lemonade stand. You decide to divide and conquer, so your friend goes to the neighborhood basketball court while you stay on your family's street corner. At the end of the day, you pool your money. Together, you've made $200, but your friend made $50 more than you. How much money did each of you make?

Or think about basketball: Shots made outside the 3-point line are worth 3 points, baskets made inside the 3-point line are worth 2 points and free throws are only worth 1 point. Your opponent is 19 points ahead of you. What combinations of baskets could you make in order to catch up?

Graphing is one of the simplest ways to solve systems of equations. All you have to do is graph both lines on the same coordinate plane, and then see where they intersect.

First, you need to write the word problem as a system of equations. Assign variables to the unknowns. Call the money you make ​Y​, and the money your friend makes ​F​.

Now you have two kinds of information: information about how much money you made together, and information about how the money you made compared to the money your friend made. Each of these will become an equation.

For the first equation, write:

\(Y + F = 200\)

since your money plus your friend's money adds up to $200.

Next, write an equation to describe the comparison between your earnings.

\(Y = F – 50\)

because the amount you made is equal to 50 dollars less than what your friend made. You could also write this equation as ​Y​ + 50 = ​F​, since what you made plus 50 dollars equals what your friend made. These are different ways of writing the same thing and will not change your final answer.

So the system of equations looks like this:

\(Y + F = 200\)
\(Y = F – 50\)

Next, you need to graph both equations on the same coordinate plane. Graph your amount, ​Y​, on the ​y​-axis and your friend's amount, ​F​, on the ​x​-axis (it actually doesn't matter which is which as long as you label them correctly). You can use graph paper and a pencil, a handheld graphing calculator or an online graphing calculator.

Right now one equation is in standard form and one is in slope-intercept form. That's not a problem, necessarily, but for the sake of consistency, get both equations into slope-intercept form.

So for the first equation, convert from standard form to slope-intercept form. That means solve for ​Y​; in other words, get ​Y​ by itself on the left side of the equals sign. So subtract ​F​ from both sides:

\(Y + F = 200\)
\(Y = -F + 200\)

Remember that in slope-intercept form, the number in front of the F is the slope and the constant is the y-intercept.

To graph the first equation, ​Y​ = −​F​ + 200, draw a point at (0, 200), and then use the slope to find more points. The slope is −1, so go down one unit and over one unit and draw a point. That creates a point at (1, 199), and if you repeat the process starting with that point, you'll get another point at (2, 198). These are tiny movements on a big line, so draw one more point at the ​x​-intercept to make sure you've got things nicely graphed in the long run. If ​Y​ = 0, then ​F​ will be 200, so draw a point at (200, 0).

To graph the second equation, ​Y​ = ​F​ – 50, use the y-intercept of −50 to draw the first point at (0, −50). Since the slope is 1, start at (0, −50), and then go up one unit and over one unit. That puts you at (1, −49). Repeat the process starting from (1, −49) and you'll get a third point at (2, −48). Again, to make sure you're doing things neatly over long distances, double-check yourself by also drawing in the ​x​-intercept. When ​Y​ = 0, ​F​ will be 50, so also draw a point at (50, 0). Draw a neat line connecting these points.

Take a close look at your graph to see where the two lines intersect. This will be the solution, because the solution to a system of equations is the point (or points) that make both equations true. On a graph, this will look like the point (or points) where the two lines intersect.

In this case, the two lines intersect at (125, 75). So the solution is that your friend (the ​x​-coordinate) made $125 and you (the ​y​-coordinate) made $75.

Quick logic check: Does this make sense? Together, the two values add to 200, and 125 is 50 more than 75. Sounds good.

In this case, there was exactly one point where the two lines crossed. When you're working with systems of equations, there are three possible outcomes, and each will look different on a graph.

•If the system has one solution, the lines will cross at a single point, as they did in the example. 
•If the system has no solutions, the lines will never cross. They will be parallel, which in algebraic terms means they will have the same slope.
•The system can also have infinite solutions, which means your "two" lines are actually the same line. So they'll have every single point in common, which is an infinite number of solutions.