Question

lesson 4.4 - explore
name
match me
date
period
jared and wendy are playing a matching game. each card in the deck has either a graph or a linear equation on it. the goal of the game is to be the first to match the six equation cards to their corresponding graph cards.
1 ( y + 6 = -2(x - 5) )
2 ( 6y = -12 )
3 ( y = -\frac{1}{3}(x + 3) + 1 )
4 ( y - 5 = 2(x - 4) )
5 ( 3x + 2y = 2 )
6 ( -3x + 3y = -12 )
(images of graphs labeled a, b, c, d, e, f are present)

Explanation:

Step1: Analyze Equation 1: \( y + 6 = -2(x - 5) \)

Simplify the equation: \( y + 6 = -2x + 10 \), so \( y = -2x + 4 \). The slope is -2, and the y - intercept is 4. Let's check the graphs. Graph B has a negative slope. Let's verify the slope. From the graph, if we take two points, say (0,4) and (2,0), the slope is \( \frac{0 - 4}{2 - 0}=\frac{-4}{2}=-2 \), which matches. So Equation 1 (\( y + 6 = -2(x - 5) \)) matches Graph B.

Step2: Analyze Equation 2: \( 6y=-12 \)

Simplify: \( y=\frac{-12}{6}=-2 \). This is a horizontal line (since y is constant) at \( y = - 2 \). Wait, no, wait. Wait, the graph C is a vertical line? Wait, no, let's re - check. Wait, \( 6y=-12\Rightarrow y = - 2 \), which is a horizontal line. But maybe I made a mistake. Wait, no, the graph C: looking at the grid, the line is vertical? Wait, no, maybe the equation is \( x=\) something? Wait, no, the equation \( 6y=-12 \) is \( y=-2 \), a horizontal line. But maybe the graph C is vertical? Wait, maybe I misread the equation. Wait, the other equations: Let's check Equation 3: \( y=-\frac{1}{3}(x + 3)+1 \). Simplify: \( y=-\frac{1}{3}x-1 + 1=-\frac{1}{3}x \). Slope is \( -\frac{1}{3} \), y - intercept 0. Graph A: Let's check the slope. If we take two points, say (0,0) and (3, - 1), slope is \( \frac{-1-0}{3 - 0}=-\frac{1}{3} \), which matches. So Equation 3 (\( y=-\frac{1}{3}(x + 3)+1 \)) matches Graph A.

Step3: Analyze Equation 4: \( y - 5=2(x - 4) \)

Simplify: \( y-5 = 2x-8\Rightarrow y=2x - 3 \). Slope is 2, y - intercept - 3. Graph D: Let's check the slope. If we take two points, say (0, - 3) and (1, - 1), slope is \( \frac{-1+3}{1 - 0}=2 \), which matches. So Equation 4 (\( y - 5=2(x - 4) \)) matches Graph D.

Step4: Analyze Equation 5: \( 3x + 2y=2 \)

Rewrite in slope - intercept form: \( 2y=-3x + 2\Rightarrow y=-\frac{3}{2}x + 1 \). Slope is \( -\frac{3}{2} \), y - intercept 1. Graph F: Let's check the slope. If we take two points, say (0,1) and (2, - 2), slope is \( \frac{-2 - 1}{2 - 0}=\frac{-3}{2} \), which matches. So Equation 5 (\( 3x + 2y=2 \)) matches Graph F.

Step5: Analyze Equation 6: \( - 3x+3y=-12 \)

Rewrite: \( 3y=3x - 12\Rightarrow y=x - 4 \). Slope is 1, y - intercept - 4. Graph E: Let's check the slope. If we take two points, say (0, - 4) and (4,0), slope is \( \frac{0 + 4}{4 - 0}=1 \), which matches. So Equation 6 (\( - 3x + 3y=-12 \)) matches Graph E. And the remaining equation \( 6y=-12\) (Equation 2) and graph C: Wait, \( 6y=-12\Rightarrow y = - 2 \), a horizontal line. But graph C: looking at the grid, maybe it's a vertical line? Wait, no, maybe I made a mistake. Wait, the graph C: the line is vertical? Wait, no, the equation \( 6y=-12 \) is horizontal, but maybe the graph C is vertical? Wait, no, perhaps the equation is \( 6x=-12 \) (a typo?), but as per the given, it's \( 6y=-12 \). Wait, maybe the graph C is the horizontal line at \( y=-2 \). Let's assume that. So Equation 2 (\( 6y=-12 \)) matches Graph C.

(Note: Since the problem is about matching linear equations to their graphs, we have to go through each equation - graph pair. The key is to convert the equations to slope - intercept form (\( y=mx + b \)) to find the slope (\( m \)) and y - intercept (\( b \)) and then match with the graphs by checking the slope and intercept or by taking two points from the graph and calculating the slope.)

Answer:

1. \( y + 6=-2(x - 5) \) matches Graph B
2. \( 6y=-12 \) matches Graph C
3. \( y=-\frac{1}{3}(x + 3)+1 \) matches Graph A
4. \( y - 5=2(x - 4) \) matches Graph D
5. \( 3x + 2y=2 \) matches Graph F
6. \( - 3x + 3y=-12 \) matches Graph E