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1. BEATING THE SYSTEM (OF
EQUATIONS)
Tackle the Math
Solve Systems of Linear Equations

2. System of Equations
■ You probably already know how to solve an equation for a variable.
■ But what about two variables? What about more than two?
■ Solving a “system of linear equations” means finding a common solution for the
equations for all variables. For example:
– 2𝑥 + 5𝑦 = 19
– 𝑥 − 2𝑦 = −4
– These equations will have specific values for x and y which they are both true.
– Generally, systems of linear equation have infinite, zero, or one solution.
■ One can use multiple methods to achieve this:
– Graphically
– Elimination
– Substitution.

3. Graphing
■ Suppose we have the following system:
– 2𝑥 + 5𝑦 = 19
– 𝑥 − 2𝑦 = −4
■ To find the x,y-values for which these equations are true, simply graph these
equations (hint: solve both sides for y first).
2𝑥 + 5𝑦 = 19 𝑥 − 2𝑦 = −4
5𝑦 = 19 − 2𝑥 −2𝑦 = −4 − 𝑥
𝑦 =
19
5
−
2
5
𝑥 𝑦 = 2 +
1
2
𝑥

4. Graphing (continued)
1. Press Y=. Enter 𝑦 =
19
5
−
2
5
𝑥 and 𝑦 = 2 +
1
2
𝑥 as functions 1
and 2, respectively.
2. Press GRAPH. You will now see the linear equations graphed.
They intersect. But where, exactly?
3. Press 2ND and TRACE. Select “5:Intersect”.
4. Press ENTER when you have the first curve selected and
ENTER again when you’ve selected the second curve. When
it says, “Guess?” move the cursor as close to the intersection
as possible and press ENTER.
5. The calculator now displays the x- and y-coordinates for the
intersection.
6. 𝑥 = 2, 𝑦 = 3. If we plug these values back into the original
equations, we see they are correct.

5. 6 + 𝑦 = 8
Pick one equation to plug in x
𝑦 = 2
2𝑥 + 0𝑦 = 12
2𝑥 = 12
𝑥 = 6
Elimination Method
■ Elimination method involves eliminating a variable
■ This can be accomplished by adding equations with a variable that has opposite
coefficients.
– E.g. 𝑥 + 𝑦 = 8 and 𝑥 − 𝑦 = 4
– The coefficients are 1 and -1 for the y variable
■ To eliminate the variable, add the equations by the “=“ sign.
𝑥 + 𝑦 = 8
𝑥 − 𝑦 = 4
Add left sides Add right sides
Thus, 𝑥 = 6 and 𝑦 = 2 is the solution to this system.

6. ■ This method works for any system of linear equations.
■ We can multiply entire equations be a nonzero constant to allow elimination.
– E.g. 2𝑥 + 3𝑦 = 7 and 𝑥 − 𝑦 = 1
■ To solve, we can multiply the 2nd equation by 3 to get opposite coefficients for
the y variable.
■ Could we multiply the 2nd equation by any other constant?
2 − 𝑦 = 1
Pick one equation to plug in x
−𝑦 = −1
5𝑥 + 0𝑦 = 10
5𝑥 = 10
𝑥 = 2
More Elimination Method
2𝑥 + 3𝑦 = 7
3𝑥 − 3𝑦 = 3
Add left sides Add right sides 𝑦 = 1
Thus, 𝑥 = 2 and 𝑦 = 1 is the solution to this system.

7. The first equation is easier
𝑥 − 2𝑦 = 2
𝑥 = 2𝑦 + 2
Plug into second equation
3(2𝑦 + 2) − 4𝑦 = 12
6𝑦 + 6 − 4𝑦 = 12
2𝑦 = 6
𝑦 = 3
Substitution Method
■ Substitution method involves substituting one equation in for the other.
– Solve one equation for a variable
– Plug first equation into the second
– Solve for first variable
– Plug first variable into either equation to find second variable
■ E.g. Solve the system: 𝑥 − 2𝑦 = 2 and 3𝑥 − 4𝑦 = 12
Thus, 𝑥 = 8 and 𝑦 = 3 is the solution to this system.
Plug y into first equation
𝑥 = 2(3) + 2
𝑥 = 6 + 2
𝑥 = 8

8. Understanding the Solutions
■ So far, we have only looked at examples that result in one solution.
– E.g. 𝑥 = −2 and 𝑦 = 3
■ If we are using substitution or elimination and receive:
– 3=17 or a=b
■ There are no (zero) solutions
■ These lines are parallel and have different y-intercepts
– 5=5 or c=c
■ There are infinite solutions
■ The lines are the same exact line (same slope and y-intercept)

9. Your Turn
Solve the following systems of equations.
2𝑥 + 3𝑦 = −8
4𝑥 − 𝑦 = −2
7𝑥 + 4𝑦 = 17
−14𝑥 − 8𝑦 = 4

10. UPCOMING
EVENTS
■ Tackle the Math Series
– Why Does Order Matter?
– 4 Out of 3 People Struggle
with Math
– Probably Probability
– Do You Know the Line?
– Beating the System (of
Equations)

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