go
topic: exploring linear equations
9. what value of k in the equation ( kx + 10 = 6y ) would give a line with slope -3?
10. what value of k in the equation ( kx - 12 = -15y ) would give a line with slope ( \frac{2}{5} )?
11. the standard form of a linear equation is ( ax + by = c ). rewrite this equation in slope - intercept form. what is the slope? what is the y - intercept?
12. if b is the y - intercept of a linear function whose graph has slope m, then ( y = mx + b ) describes the line. below is an incomplete justification of this statement. fill in the missing information.
| statements | reasons |
| --- | --- |
| 1. ( m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}} ) | 1. slope formula |
| 2. ( m=\frac{y - b}{x - 0} ) | 2. by definition, if b is the y - intercept, then (, b) is a point on the line. (x, y) is any other point on the line. |
| 3. ( m=\frac{y - b}{x} ) | 3.? |
| 4. ( mquad = y - b ) | 4. multiplication property of equality (multiply both sides of the equation by x.) |
| 5. ( mx + b = y ), or ( y = mx + b ) | 5.? |

Question

Accepted Answer

### Question 9
# Explanation:
## Step1: Rewrite in slope - intercept form
We know that the slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.
Start with the equation $kx + 10=6y$.
Divide both sides of the equation by 6: $y=\frac{k}{6}x+\frac{10}{6}$.
Simplify $\frac{10}{6}$ to $\frac{5}{3}$, so $y = \frac{k}{6}x+\frac{5}{3}$.

## Step2: Set slope equal to - 3
The slope $m=\frac{k}{6}$, and we want the slope to be - 3.
So we set up the equation $\frac{k}{6}=-3$.

## Step3: Solve for k
Multiply both sides of the equation $\frac{k}{6}=-3$ by 6: $k=-3	imes6=-18$.

# Answer:
The value of $k$ is $- 18$.

### Question 10
# Explanation:
## Step1: Rewrite in slope - intercept form
The slope - intercept form is $y = mx + b$.
Start with the equation $kx-12 = - 15y$.
First, add 12 to both sides: $kx=-15y + 12$.
Then, divide both sides by - 15: $y=-\frac{k}{15}x-\frac{12}{15}$.
Simplify $-\frac{12}{15}$ to $-\frac{4}{5}$, so $y=-\frac{k}{15}x-\frac{4}{5}$.

## Step2: Set slope equal to $\frac{2}{5}$
The slope $m =-\frac{k}{15}$, and we want $m=\frac{2}{5}$.
Set up the equation $-\frac{k}{15}=\frac{2}{5}$.

## Step3: Solve for k
Multiply both sides of the equation $-\frac{k}{15}=\frac{2}{5}$ by - 15:
$k=\frac{2}{5}	imes(-15)=2	imes(-3)=-6$.

# Answer:
The value of $k$ is $-6$.

### Question 11
# Explanation:
## Step1: Rewrite $Ax + By=C$ in slope - intercept form
We want to solve the equation $Ax + By = C$ for $y$.
Subtract $Ax$ from both sides: $By=-Ax + C$.
Divide both sides by $B$ (assuming $B
eq0$): $y=-\frac{A}{B}x+\frac{C}{B}$.

## Step2: Identify the slope
In the slope - intercept form $y = mx + b$, the coefficient of $x$ is the slope.
For $y=-\frac{A}{B}x+\frac{C}{B}$, the slope $m =-\frac{A}{B}$.

## Step3: Identify the y - intercept
In the slope - intercept form $y=mx + b$, the constant term is the y - intercept.
For $y =-\frac{A}{B}x+\frac{C}{B}$, the y - intercept $b=\frac{C}{B}$.

# Answer:
- Slope - intercept form: $y =-\frac{A}{B}x+\frac{C}{B}$ (when $B
eq0$)
- Slope: $-\frac{A}{B}$
- y - intercept: $\frac{C}{B}$

### Question 12
## Statement 2:
By definition, if $b$ is the y - intercept, then the point $(0,b)$ is on the line (because the y - intercept is the point where $x = 0$). So in the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$, we can take $(x_1,y_1)=(0,b)$ and $(x_2,y_2)=(x,y)$. Then $m=\frac{y - b}{x-0}$.

## Reason 3:
Simplify the denominator $x - 0$ to $x$. So the reason for $m=\frac{y - b}{x}$ (from $m=\frac{y - b}{x - 0}$) is "Simplify $x-0$ to $x$" (or "Simplification of the denominator").

## Statement 4:
From $m=\frac{y - b}{x}$, when we multiply both sides by $x$, we get $mx=y - b$.

## Reason 5:
From $mx=y - b$, we add $b$ to both sides of the equation (Addition Property of Equality) to get $mx + b=y$, or $y=mx + b$.

# Brief Explanations:
- Statement 2: The y - intercept occurs at $x = 0$, so the point is $(0,b)$.
- Reason 3: Simplify $x-0$ to $x$.
- Statement 4: $mx=y - b$ (after multiplying both sides of $m=\frac{y - b}{x}$ by $x$)
- Reason 5: Use the Addition Property of Equality (add $b$ to both sides of $mx=y - b$) to get $y=mx + b$.

# Answer:
- Statement 2: The point is $(0,b)$ (so the first blank in the point $(,b)$ is 0)
- Reason 3: Simplify $x - 0$ to $x$ (or "Simplification of the denominator")
- Statement 4: $mx=y - b$
- Reason 5: Addition Property of Equality (add $b$ to both sides of $mx=y - b$)

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 9

Step1: Rewrite in slope - intercept form

Step2: Set slope equal to - 3

Step3: Solve for k

Step1: Rewrite in slope - intercept form

Step2: Set slope equal to $\frac{2}{5}$

Step3: Solve for k

Step1: Rewrite $Ax + By=C$ in slope - intercept form

Step2: Identify the slope

Step3: Identify the y - intercept

Answer:

Question 10

statements	reasons
2. ( m=\frac{y - b}{x - 0} )	2. by definition, if b is the y - intercept, then (, b) is a point on the line. (x, y) is any other point on the line.
3. ( m=\frac{y - b}{x} )	3.?
4. ( mquad = y - b )	4. multiplication property of equality (multiply both sides of the equation by x.)
5. ( mx + b = y ), or ( y = mx + b )	5.?