Question

a. standard form → slope-intercept form
rewrite each equation in slope-intercept form $y = mx + b$.

1. $2x + y = 7$
2. $3x - 2y = 10$
3. $5y + 4x = 20$
4. $6x - 3y = -12$

b. slope-intercept form → standard form
rewrite each equation in standard form $ax + by = c$ where $a$ is positive.

1. $y = 2x - 5$
2. $y = -\frac{1}{3}x + 4$
3. $y = \frac{5}{2}x - 1$

c. point-slope form → slope-intercept form
rewrite each equation in slope-intercept form.

1. $y - 3 = 2(x + 1)$
2. $y + 4 = -\frac{1}{2}(x - 6)$
3. $y - 1 = 5(x - 2)$

Explanation:

Part A: Standard Form → Slope - Intercept Form

1. \(2x + y=7\)

Step 1: Isolate \(y\)

Subtract \(2x\) from both sides of the equation.
\(y=- 2x + 7\)

2. \(3x-2y = 10\)

Step 1: Isolate the \(y\) - term

Subtract \(3x\) from both sides: \(-2y=-3x + 10\)

Step 2: Solve for \(y\)

Divide each term by \(-2\): \(y=\frac{3}{2}x-5\)

3. \(5y + 4x=20\)

Step 1: Isolate the \(y\) - term

Subtract \(4x\) from both sides: \(5y=-4x + 20\)

Step 2: Solve for \(y\)

Divide each term by \(5\): \(y=-\frac{4}{5}x + 4\)

4. \(6x-3y=-12\)

Step 1: Isolate the \(y\) - term

Subtract \(6x\) from both sides: \(-3y=-6x-12\)

Step 2: Solve for \(y\)

Divide each term by \(-3\): \(y = 2x+4\)

Part B: Slope - Intercept Form → Standard Form

5. \(y = 2x-5\)

Step 1: Move \(x\) - term to the left

Subtract \(2x\) from both sides: \(-2x + y=-5\)

Step 2: Make \(A\) positive

Multiply each term by \(-1\): \(2x-y = 5\)

6. \(y=-\frac{1}{3}x + 4\)

Step 1: Move \(x\) - term to the left

Add \(\frac{1}{3}x\) to both sides: \(\frac{1}{3}x + y=4\)

Step 2: Eliminate fraction

Multiply each term by \(3\): \(x + 3y=12\)

7. \(y=\frac{5}{2}x-1\)

Step 1: Move \(x\) - term to the left

Subtract \(\frac{5}{2}x\) from both sides: \(-\frac{5}{2}x + y=-1\)

Step 2: Eliminate fraction

Multiply each term by \(2\): \(-5x + 2y=-2\)

Step 3: Make \(A\) positive

Multiply each term by \(-1\): \(5x-2y = 2\)

Part C: Point - Slope Form → Slope - Intercept Form

8. \(y - 3=2(x + 1)\)

Step 1: Distribute the slope

\(y-3 = 2x+2\)

Step 2: Solve for \(y\)

Add \(3\) to both sides: \(y=2x + 5\)

9. \(y + 4=-\frac{1}{2}(x - 6)\)

Step 1: Distribute the slope

\(y + 4=-\frac{1}{2}x+3\)

Step 2: Solve for \(y\)

Subtract \(4\) from both sides: \(y=-\frac{1}{2}x-1\)

10. \(y - 1=5(x - 2)\)

Step 1: Distribute the slope

\(y-1 = 5x-10\)

Step 2: Solve for \(y\)

Add \(1\) to both sides: \(y=5x-9\)

Answer:

s

Part A

1. \(y=-2x + 7\)
2. \(y=\frac{3}{2}x-5\)
3. \(y=-\frac{4}{5}x + 4\)
4. \(y = 2x+4\)

Part B

1. \(2x-y = 5\)
2. \(x + 3y=12\)
3. \(5x-2y = 2\)

Part C

1. \(y=2x + 5\)
2. \(y=-\frac{1}{2}x-1\)
3. \(y=5x-9\)