Question

1. which equation represents the line whose slope is $\frac{1}{2}$ and whose y-intercept is 5?a. $y = \frac{1}{2}x + 5$b. $y = 5x + \frac{1}{2}$c. $y = \frac{1}{2}x - 5$d. $y = 5x - \frac{1}{2}$2. write an equation of the line whose slope is 2 and whose y-intercept is $-3$.3. write an equation of the straight line whose slope is 2 and whose y-intercept is the same as that of the line represented by the equation $y = 4x - 2$.4. which is an equation of the line that passes through the point (1,4) and has a slope of 3?a. $y = 3x + 4$b. $y = \frac{1}{3}x + 4$c. $y = 3x - 1$d. $y = 3x + 1$5. write an equation of the line that passes through the point (0,3) and whose slope is 2.6. the graph of which equation passes through points (0,6) and (4,$-1$)?a. $y = \frac{7}{4}x + 6$b. $y = \frac{4}{7}x + 6$c. $y = -\frac{7}{4}x + 6$d. $y = -\frac{4}{7}x + 6$

Explanation:

All problems use the slope-intercept form $y = mx + b$, where $m$ = slope, $b$ = y-intercept, or point-slope form $y - y_1 = m(x - x_1)$ for points on the line.

Problem 1

Step1: Recall slope-intercept form

$y = mx + b$

Step2: Substitute $m=\frac{1}{2}, b=5$

$y = \frac{1}{2}x + 5$

Problem 2

Step1: Recall slope-intercept form

$y = mx + b$

Step2: Substitute $m=2, b=-3$

$y = 2x - 3$

Problem 3

Step1: Identify y-intercept of $y=4x-2$

$b = -2$

Step2: Substitute $m=2, b=-2$

$y = 2x - 2$

Problem 4

Step1: Use point-slope form

$y - 4 = 3(x - 1)$

Step2: Simplify to slope-intercept form

$y = 3x - 3 + 4 = 3x + 1$

Problem 5

Step1: Identify $b$ from $(0,3)$

$b = 3$

Step2: Substitute $m=2, b=3$

$y = 2x + 3$

Problem 6

Step1: Calculate slope $m$

$m = \frac{-1 - 6}{4 - 0} = -\frac{7}{4}$

Step2: Identify $b$ from $(0,6)$

$b = 6$

Step3: Substitute into slope-intercept form

$y = -\frac{7}{4}x + 6$

Answer:

1. A. $y = \frac{1}{2}x + 5$
2. $y = 2x - 3$
3. $y = 2x - 2$
4. D. $y = 3x + 1$
5. $y = 2x + 3$
6. C. $y = -\frac{7}{4}x + 6$