Lesson 55: Solving Systems of Linear Equations by Graphing

New Concept

A system of linear equations consists of two or more linear equations containing two or more variables.

What’s next

Next, you’ll graph these equations on a coordinate plane to visually find the one point that solves the entire system.

Property

A solution of a system of linear equations is any ordered pair that makes all the equations true.

Explanation

Think of a solution as a secret password! An ordered pair $(x, y)$ is the correct password only if it works for every single equation in the system. If it fails even one, the whole thing is wrong. To check, substitute the x and y values into each equation. If they all result in true statements, you've cracked the code!

Examples

• Is $(2, 1)$ a solution for $2x + y = 5$ and $x - y = 1$? Check: $2(2) + 1 = 5$ ,$2-1=1$
• Is $(4, 1)$ a solution for $x + y = 5$ and $x - 2y = 3$? Check: $4 + 1 = 5$ ,$4-2(1)=3$
• Is $(-1, 3)$ a solution for $y = -2x + 1$ and $y = x + 4$? Check: $3 = -2(-1) + 1$ ,$3=-1+4$

A point might solve one equation, but does it solve the whole system? Let's check. This card highlights the key idea of verifying if an ordered pair is a solution to a system of equations.

Example Problem
Tell whether the ordered pair $(5, 1)$ is a solution of the system: $2x + 3y = 13$, $x = 8 - 2y$.

Step-by-Step

1. To verify, we substitute $5$ for $x$ and $1$ for $y$ in each equation and check if the statements are true.
2. We check the first equation, $2x + 3y = 13$:

The ordered pair works in the first equation.

3. Now we check the second equation, $x = 8 - 2y$:

The ordered pair does not work in the second equation.

4. Since the ordered pair $(5, 1)$ only makes one equation true, it is not a solution to the entire system.

Property

If a system of linear equations has one solution, the solution is the common point or the point of intersection of their graphs.

Explanation

Solving a system by graphing is like a treasure hunt on a map! Each equation is a line showing a path. The solution is the treasure, and it's buried at the exact spot where the two lines cross. All you have to do is graph both lines on the same coordinate plane and find their point of intersection, $(x, y)$.

Examples

• To solve the system $y = x + 1$ and $y = -x + 5$, graph both lines. You'll see they intersect at the point $(2, 3)$, which is the solution.
• Solve the system $y = 2x$ and $x + y = 6$. Rewrite the second equation as $y = -x + 6$. Graphing both lines reveals they intersect at the solution, $(2, 4)$.
• Check the solution $(2, 4)$ for $y = 2x$ and $y = -x + 6$. Substitute into both: $4 = 2(2)

\rightarrow 4=4 \text{(True)}$and$4 = -2 + 6
\rightarrow 4=4 \text{(True)}$. The intersection point is correct!

Equations often come in disguise. Let's unveil their true linear form to find where they meet. This example card demonstrates the key idea of solving a system by graphing, including rearranging equations first.

Example Problem
Solve the system by graphing and check your solution: $2x + y = 7$, $x - y = -1$.

Step-by-Step

1. To solve by graphing, we first need to rewrite both equations into slope-intercept form, $y = mx + b$.
2. Rearrange the first equation, $2x + y = 7$:

This line has a y-intercept of $7$ and a slope of $-2$.

3. Rearrange the second equation, $x - y = -1$:

This line has a y-intercept of $1$ and a slope of $1$.

4. Graph both equations. The line $y = -2x + 7$ starts at $y=7$ and moves down 2 units for every 1 unit to the right. The line $y = x + 1$ starts at $y=1$ and moves up 1 unit for every 1 unit to the right. The lines intersect at the point $(2, 3)$.
5. Check that the ordered pair $(2, 3)$ makes both original equations true.

6. The ordered pair $(2, 3)$ is a solution to both equations, so it is the solution to the system.

Property

Create a system of equations where each equation represents a different rate plan. The variables typically stand for cost $(y)$ and a unit of time or quantity $(x)$. The solution $(x, y)$ is the break-even point where both plans cost the same.

Explanation

Is the expensive gym with no sign-up fee a better deal than the cheap gym with a big fee? Systems of equations can tell you! By turning each option into an equation and graphing them, the intersection point shows the exact moment their costs are equal. This helps you figure out which plan saves you more money in the long run.

Examples

• Phone Plan A costs 20 dollars per month plus 10 dollars per gigabyte: $y = 10x + 20$. Plan B is 40 dollars per month plus 5 dollars per gigabyte: $y = 5x + 40$. The lines intersect at $(4, 60)$, meaning at 4 gigabytes, both plans cost 60 dollars.
• Car Rental A is 50 dollars plus 1 dollar per mile: $y = x + 50$. Car Rental B is 2 dollars per mile: $y = 2x$. They intersect at $(50, 100)$. At 50 miles, both rentals cost 100 dollars.