Lesson 6 Extra Practice Equations In Y Mx B Form Answers - Fill ... - Free Printable

Problem Overview:

The task involves writing equations of lines in slope-intercept form ($y = mx + b$), where:
- $m$ is the slope,
- $b$ is the y-intercept.

We are given specific slopes and y-intercepts for some problems, and for others, we need to determine the slope and y-intercept from tables or graphs. Let's solve each part step by step.

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Part 1: Write an equation of a line in slope-intercept form with the given slope and y-intercept

#### 1. Slope: 3, y-intercept: -5
- Slope ($m$) = 3
- y-intercept ($b$) = -5
- Equation: $y = mx + b$
- Substituting values: $y = 3x - 5$

Answer: $y = 3x - 5$

#### 2. Slope: 2, y-intercept: -6
- Slope ($m$) = 2
- y-intercept ($b$) = -6
- Equation: $y = mx + b$
- Substituting values: $y = 2x - 6$

Answer: $y = 2x - 6$

#### 3. Slope: -6, y-intercept: $\frac{1}{2}$
- Slope ($m$) = -6
- y-intercept ($b$) = $\frac{1}{2}$
- Equation: $y = mx + b$
- Substituting values: $y = -6x + \frac{1}{2}$

Answer: $y = -6x + \frac{1}{2}$

#### 4. Slope: -7, y-intercept: $\frac{3}{4}$
- Slope ($m$) = -7
- y-intercept ($b$) = $\frac{3}{4}$
- Equation: $y = mx + b$
- Substituting values: $y = -7x + \frac{3}{4}$

Answer: $y = -7x + \frac{3}{4}$

#### 5. Slope: $\frac{1}{2}$, y-intercept: 7
- Slope ($m$) = $\frac{1}{2}$
- y-intercept ($b$) = 7
- Equation: $y = mx + b$
- Substituting values: $y = \frac{1}{2}x + 7$

Answer: $y = \frac{1}{2}x + 7$

#### 6. Slope: $\frac{3}{4}$, y-intercept: -8
- Slope ($m$) = $\frac{3}{4}$
- y-intercept ($b$) = -8
- Equation: $y = mx + b$
- Substituting values: $y = \frac{3}{4}x - 8$

Answer: $y = \frac{3}{4}x - 8$

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Part 2: Write an equation in slope-intercept form for each table

#### 7. Table:
| $x$ | 0 | 1 | 2 |
|-----|---|---|---|
| $y$ | 0 | 1 | 2 |

- Step 1: Identify the y-intercept ($b$).
- When $x = 0$, $y = 0$. So, the y-intercept is $b = 0$.

- Step 2: Calculate the slope ($m$).
- Slope formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
- Using points $(0, 0)$ and $(1, 1)$:
$$
m = \frac{1 - 0}{1 - 0} = 1
$$

- Step 3: Write the equation.
- Slope ($m$) = 1
- y-intercept ($b$) = 0
- Equation: $y = mx + b$
- Substituting values: $y = 1x + 0$ or $y = x$

Answer: $y = x$

#### 8. Table:
| $x$ | -1 | 0 | 1 |
|-----|----|---|---|
| $y$ | -3 | -1 | 1 |

- Step 1: Identify the y-intercept ($b$).
- When $x = 0$, $y = -1$. So, the y-intercept is $b = -1$.

- Step 2: Calculate the slope ($m$).
- Slope formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
- Using points $(0, -1)$ and $(1, 1)$:
$$
m = \frac{1 - (-1)}{1 - 0} = \frac{1 + 1}{1} = 2
$$

- Step 3: Write the equation.
- Slope ($m$) = 2
- y-intercept ($b$) = -1
- Equation: $y = mx + b$
- Substituting values: $y = 2x - 1$

Answer: $y = 2x - 1$

#### 9. Table:
| $x$ | 2 | 3 | 4 |
|-----|---|---|---|
| $y$ | 6 | 9 | 12 |

- Step 1: Identify the y-intercept ($b$).
- To find the y-intercept, we need to determine the value of $y$ when $x = 0$.
- From the pattern, observe that $y$ increases by 3 for every increase of 1 in $x$. This suggests a linear relationship with a slope of 3.
- Using the point $(2, 6)$:
$$
y = mx + b \implies 6 = 3(2) + b \implies 6 = 6 + b \implies b = 0
$$

- Step 2: Confirm the slope ($m$).
- Slope formula: $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$
- Using points $(2, 6)$ and $(3, 9)$:
$$
m = \frac{9 - 6}{3 - 2} = \frac{3}{1} = 3
$$

- Step 3: Write the equation.
- Slope ($m$) = 3
- y-intercept ($b$) = 0
- Equation: $y = mx + b$
- Substituting values: $y = 3x$

Answer: $y = 3x$

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Part 3: Write an equation in slope-intercept form for each graph

#### 10. Graph:
- The graph shows a line passing through the origin (0, 0) with a slope of 1.
- Step 1: Identify the y-intercept ($b$).
- The line passes through the origin, so the y-intercept is $b = 0$.

- Step 2: Identify the slope ($m$).
- The line rises 1 unit for every 1 unit it runs to the right, so the slope is $m = 1$.

- Step 3: Write the equation.
- Slope ($m$) = 1
- y-intercept ($b$) = 0
- Equation: $y = mx + b$
- Substituting values: $y = 1x + 0$ or $y = x$

Answer: $y = x$

#### 11. Graph:
- The graph shows a line with a negative slope and a y-intercept at (0, 2).
- Step 1: Identify the y-intercept ($b$).
- The line crosses the y-axis at $y = 2$, so the y-intercept is $b = 2$.

- Step 2: Identify the slope ($m$).
- The line falls 1 unit for every 1 unit it runs to the right, so the slope is $m = -1$.

- Step 3: Write the equation.
- Slope ($m$) = -1
- y-intercept ($b$) = 2
- Equation: $y = mx + b$
- Substituting values: $y = -1x + 2$ or $y = -x + 2$

Answer: $y = -x + 2$

#### 12. Graph:
- The graph shows a horizontal line passing through $y = 3$.
- Step 1: Identify the y-intercept ($b$).
- The line is horizontal and passes through $y = 3$, so the y-intercept is $b = 3$.

- Step 2: Identify the slope ($m$).
- A horizontal line has a slope of 0.

- Step 3: Write the equation.
- Slope ($m$) = 0
- y-intercept ($b$) = 3
- Equation: $y = mx + b$
- Substituting values: $y = 0x + 3$ or $y = 3$

Answer: $y = 3$

#### 13. Graph:
- The graph shows a line with a positive slope and a y-intercept at (0, -2).
- Step 1: Identify the y-intercept ($b$).
- The line crosses the y-axis at $y = -2$, so the y-intercept is $b = -2$.

- Step 2: Identify the slope ($m$).
- The line rises 2 units for every 1 unit it runs to the right, so the slope is $m = 2$.

- Step 3: Write the equation.
- Slope ($m$) = 2
- y-intercept ($b$) = -2
- Equation: $y = mx + b$
- Substituting values: $y = 2x - 2$

Answer: $y = 2x - 2$

#### 14. Graph:
- The graph shows a line with a positive slope and a y-intercept at (0, 1).
- Step 1: Identify the y-intercept ($b$).
- The line crosses the y-axis at $y = 1$, so the y-intercept is $b = 1$.

- Step 2: Identify the slope ($m$).
- The line rises 1 unit for every 1 unit it runs to the right, so the slope is $m = 1$.

- Step 3: Write the equation.
- Slope ($m$) = 1
- y-intercept ($b$) = 1
- Equation: $y = mx + b$
- Substituting values: $y = 1x + 1$ or $y = x + 1$

Answer: $y = x + 1$

#### 15. Graph:
- The graph shows a line with a positive slope and a y-intercept at (0, 0).
- Step 1: Identify the y-intercept ($b$).
- The line passes through the origin, so the y-intercept is $b = 0$.

- Step 2: Identify the slope ($m$).
- The line rises 2 units for every 1 unit it runs to the right, so the slope is $m = 2$.

- Step 3: Write the equation.
- Slope ($m$) = 2
- y-intercept ($b$) = 0
- Equation: $y = mx + b$
- Substituting values: $y = 2x$

Answer: $y = 2x$

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Final Answers:

1. $y = 3x - 5$
2. $y = 2x - 6$
3. $y = -6x + \frac{1}{2}$
4. $y = -7x + \frac{3}{4}$
5. $y = \frac{1}{2}x + 7$
6. $y = \frac{3}{4}x - 8$
7. $y = x$
8. $y = 2x - 1$
9. $y = 3x$
10. $y = x$
11. $y = -x + 2$
12. $y = 3$
13. $y = 2x - 2$
14. $y = x + 1$
15. $y = 2x$

$$
\boxed{
\begin{aligned}
&1. y = 3x - 5 \\
&2. y = 2x - 6 \\
&3. y = -6x + \frac{1}{2} \\
&4. y = -7x + \frac{3}{4} \\
&5. y = \frac{1}{2}x + 7 \\
&6. y = \frac{3}{4}x - 8 \\
&7. y = x \\
&8. y = 2x - 1 \\
&9. y = 3x \\
&10. y = x \\
&11. y = -x + 2 \\
&12. y = 3 \\
&13. y = 2x - 2 \\
&14. y = x + 1 \\
&15. y = 2x
\end{aligned}
}
$$