Question

graphing vs. substitution
1
$y = x + 6$
$y = -2x - 3$
$y + 6 = -2x - 3$
(graph grid)
$y = x + 6$
$y = -2x - 3$
2
$5x - y = -5$
$3x - 6y = 24$
(handwritten notes)
(graph grid with lines)
$5x - y = -5$
$3x - 6y = 24$
(handwritten notes with green highlights)
3
$2x - 3y = -12$
$x + y = 9$
(graph grid)
$2x - 3y = -12$
$x + y = 9$
(handwritten notes with green highlight)
4
$x - 3y = -15$
$x = -3$
(handwritten notes)
(graph grid)
$x - 3y = -15$
$x = -3$
(handwritten notes with $y = 4$ and $(-3, 4)$)
5
$y = 2x - 4$
$6x - 3y = 12$
(graph grid)
$y = 2x - 4$
$6x - 3y = 12$

Explanation:

Let's solve the first system of equations using substitution (we can choose any, let's pick problem 1: \( y = x + 6 \) and \( y = -2x - 3 \))

Step1: Substitute \( y \) from first equation into second

Since \( y = x + 6 \) and \( y = -2x - 3 \), we set \( x + 6 = -2x - 3 \)

Step2: Solve for \( x \)

Add \( 2x \) to both sides: \( x + 2x + 6 = -3 \) → \( 3x + 6 = -3 \)
Subtract 6 from both sides: \( 3x = -3 - 6 \) → \( 3x = -9 \)
Divide by 3: \( x = \frac{-9}{3} = -3 \)

Step3: Solve for \( y \)

Substitute \( x = -3 \) into \( y = x + 6 \): \( y = -3 + 6 = 3 \)

Answer:

The solution is \( x = -3 \), \( y = 3 \) (or the ordered pair \( (-3, 3) \))