1. Solving Linear Systems Graphically
Chapter
Solving Linear Systems Graphically

2. Graphing Substitution Elimination Vocabulary:
A linear system or system of linear equations is a set of two or more equations with the same set of unknowns.
𝑦=−𝑥+3 𝑦=𝑥+1 is a linear system whose graph is shown
The solution of a linear system is an ordered pair (𝑥,𝑦) that makes both equations true.
The three methods that can be used to solve a linear system are
Graphing
Substitution
Elimination

3. Vocabulary: The solution of a linear systems is an ordered pair that makes both equations true statements. To determine whether an ordered pair is a solution to a linear system, substitute the values of x and y into BOTH equations. Is −1,−5 a solution of 3𝑥−𝑦=2 −𝑥−𝑦=−6 ? 3 −1 − −5 =2 → −3+5=2 − −1 − −5 =−6 → 1+5≠−6 Since the second equation does not make a true statement, (−1,−5) is NOT a solution of the linear system.

4. To solve a linear system graphically, Graph each linear equation.
Vocabulary:
To solve a linear system graphically,
Graph each linear equation.
Identify the coordinates of the point where the graphs cross. This is the solution.
Check your answer by substituting the values into the original equations.
For example, solve 𝑦=−𝑥+3 𝑦=𝑥+5
Graphing each line, we see that the graphs intersect at (−1,4).
Check: 4=−(−1)+3 4=−1+5 CORRECT
The solution is (−𝟏,𝟒)
All graphs created on Desmos.com

5. When you solve a linear system by graphing, three things can happen:
Vocabulary:
When you solve a linear system by graphing, three things can happen:
The graphs intersect at exactly one point. There is ONE solution.
The graphs are parallel lines. They never intersect, therefore there is NO solution.
The graphs are both the same line. They always intersect, therefore there are INFINITELY MANY SOLUTIONS.
One Solution No Solution Infinitely Many Solutions
All graphs created on Desmos.com

6. Examples
Determine whether (2,1) is the solution of the linear system 𝑦−2=−2(𝑥−1) 𝑦−1=𝑥 Substitute (2,1) into each equation to see whether it makes a true statement. 1−2=−2 2−1 → 1−2≠−2 1−1=2 → 0≠2 𝟐,𝟏 is NOT a solution of the linear system. Determine how many solutions 𝑦=−3𝑥−2 2𝑦=−6𝑥−4 has. When we simplify the second equation to prepare to graph it, we find 2𝑦=−6𝑥−4 → 𝑦=−3𝑥−2 Since this is the same as the first equation, we know there are infinitely many solutions.

7. Examples
Solve the linear system by graphing. 𝑦=−𝑥+4 𝑦=4𝑥−1 1. Graph each equation. 𝑦=−𝑥+4 𝑦=4𝑥−1 2. Where do they cross? (1,3) 3. Check your answer. 3=−1+4 3=4(1)−1 CORRECT The solution of the linear system is (1,3)

8. Let’s Try Some Examples
1. Determine whether (2,2) is a solution of 𝑦=3𝑥−4 𝑦=𝑥 . 2. How many solutions does 𝑦=−𝑥−8 𝑦=−𝑥+11 have? How can you tell? 3. Solve 𝑦=−𝑥−4 𝑦=4𝑥−4 graphically. Check your answer.

9. Let’s Check Our Answers
1. Determine whether (2,2) is a solution of 𝑦=3𝑥−4 𝑦=𝑥 . 2=3 2 −4 2=2 CORRECT. (2,2) is a solution. 2. How many solutions does 𝑦=−𝑥−8 𝑦=−𝑥+11 have? How can you tell? No solution. The lines have the same slope, meaning they’re parallel. Since they never intersect, there are no solutions to the linear system. 3. Solve 𝑦=−𝑥−4 𝑦=4𝑥−4 graphically. Check your answer. (0,−4) is the solution of the linear system. −4=0−4 −4=4 0 −4 CORRECT