Question

fluency and skills practice
name:
lesson 9
deriving $y = mx + b$ continued
graphs
5
$y = 4x + 5$
6
$y = 40x + 20$
7
$y = 25x + 50$
8 explain one way you could know that the equation of a line is wrong without performing calculations.

Explanation:

For Question 5 (Graph 5)

Step1: Identify the y - intercept ($b$)

The y - intercept is the value of $y$ when $x = 0$. From the graph, when $x = 0$, $y=4$. So $b$ should be 4.

Step2: Calculate the slope ($m$)

We can use two points. Let's take $(0,4)$ and $(5,10)$ (from the graph). The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So $m=\frac{10 - 4}{5 - 0}=\frac{6}{5}=1.2$ (or $\frac{6}{5}$). But the student wrote $y = 4x+5$, which is wrong. The correct equation should be $y=\frac{6}{5}x + 4$ (or $y = 1.2x+4$).

For Question 6 (Graph 6)

Step1: Identify the y - intercept ($b$)

When $x = 0$, $y = 40$. So $b = 40$.

Step2: Calculate the slope ($m$)

Take two points, say $(0,40)$ and $(5,140)$ (since when $x = 5$, $y=40 + 100=140$ as the grid seems to have a rise of 40 per some x - value? Wait, actually, from the graph, the y - axis has 40,80,120,160,200,240. Let's take $(0,40)$ and $(5,240)$? No, wait, the line goes from $(0,40)$ and let's see the slope. The student wrote $y = 40x+20$, but $b$ should be 40. Let's calculate slope: $\frac{y_2 - y_1}{x_2 - x_1}=\frac{240 - 40}{10 - 0}=\frac{200}{10}=20$. So the correct equation is $y = 20x+40$.

For Question 7 (Graph 7)

Step1: Identify the y - intercept ($b$)

When $x = 0$, $y = 50$. So $b = 50$.

Step2: Calculate the slope ($m$)

Take two points, $(0,50)$ and $(4,150)$ (since when $x = 4$, $y = 50+100 = 150$? Wait, the y - axis has 50,75,100,125,150. Let's take $(0,50)$ and $(4,150)$. Then $m=\frac{150 - 50}{4 - 0}=\frac{100}{4}=25$. Wait, the student wrote $y = 25x+50$, which is correct? Wait, when $x = 4$, $y=25\times4 + 50=100 + 50=150$, which matches the graph. Wait, maybe I misread the graph. Let's check: when $x = 0$, $y = 50$ (correct, $b = 50$). Slope: from $(0,50)$ to $(4,150)$, slope is 25. So the student's answer $y = 25x+50$ is correct? Wait, maybe the graph for question 7: when $x = 0$, $y = 50$, and the slope is 25. So the equation is correct.

For Question 8

Answer:

For question 5: Correct equation is $y=\frac{6}{5}x + 4$ (or $y = 1.2x+4$)
For question 6: Correct equation is $y = 20x+40$
For question 7: The student's equation $y = 25x+50$ is correct (since $b = 50$ and slope $m = 25$ matches the graph)
For question 8: One way is to check the y - intercept (where the line crosses the y - axis, $x = 0$). If the equation's $b$ (y - intercept) doesn't match the graph's y - intercept, the equation is wrong. Also, we can check the slope's steepness visually: if the line on the graph is less steep than the slope in the equation (or more steep), it's wrong.