Math / Algebra

QuestionTopic 1: Writing Equations
1. (Multiple Choice): Write the equation of the line that passes through $(3,-2)$ with a slope of 4 . a) $y=4 x-14$ b) $y=4 x+14$ c) $y=4 x-2$ d) $y=-4 x+2$
2. (Multiple Choice): What is the slope-intercept form of a line with slope -5 and $y$ -intercept $(0,4)$ ? a) $y=-5 x+4$ b) $y=5 x+4$ c) $y=-5 x-4$ d) $y=5 x-4$
3. (Multiple Choice): Convert the equation $3 x-4 y=12$ to slope-intercept form. a) $y=3 / 4 x+3$ b) $y=3 / 4 x-3$ c) $y=4 / 3 x+12$ d) $y=-4 / 3 x-12$
4. (Free Response): Write the equation of a line parallel to $y=2 x+3$ passing through $(1,2)$ .
5. (Free Response): Write the equation of a line passing through the points $(0,4)$ and $(2,8)$ .
6. (Free Response): Find the slope of a line that passes through $(-3,7)$ and $(4,-1)$ . Finish in Slope Intercept form!
Topic 2: Parallel vs Perpendicular vs Neither
1. (Multiple Choice): Determine whether the lines $y=3 x+2$ and $y=-1 / 3 x-5$ are parallel, perpendicular, or neither. a) Parallel b) Perpendicular c) Neither

Studdy Solution

STEP 1

1. We are dealing with linear equations in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We need to understand how to manipulate equations to find slope-intercept form and determine relationships between lines.

STEP 2

1. Solve Topic 1: Writing Equations - Write the equation of a line given a point and a slope. - Identify the slope-intercept form given a slope and y-intercept. - Convert a standard form equation to slope-intercept form. - Write the equation of a line parallel to a given line through a point. - Write the equation of a line through two points. - Find the slope of a line through two points and express in slope-intercept form.
2. Solve Topic 2: Parallel vs Perpendicular vs Neither - Determine the relationship between two lines based on their slopes.

STEP 3

1.1 (Multiple Choice): Write the equation of the line that passes through $(3, -2)$ with a slope of 4.
Use the point-slope form of a line equation: $y - y_1 = m(x - x_1)$ .
Given: - Point $(x_1, y_1) = (3, -2)$ - Slope $m = 4$
Substitute into the point-slope form:
$y - (-2) = 4(x - 3)$ $y + 2 = 4x - 12$ $y = 4x - 12 - 2$ $y = 4x - 14$
The correct answer is: a) $y = 4x - 14$

STEP 4

1.2 (Multiple Choice): What is the slope-intercept form of a line with slope -5 and y-intercept $(0, 4)$ ?
The slope-intercept form is $y = mx + b$ .
Given: - Slope $m = -5$ - y-intercept $b = 4$
Substitute into the slope-intercept form:
$y = -5x + 4$
The correct answer is: a) $y = -5x + 4$

STEP 5

1.3 (Multiple Choice): Convert the equation $3x - 4y = 12$ to slope-intercept form.
Start with the given equation:
$3x - 4y = 12$
Solve for $y$ :
$-4y = -3x + 12$ $y = \frac{3}{4}x - 3$
The correct answer is: b) $y = \frac{3}{4}x - 3$

STEP 6

1.4 (Free Response): Write the equation of a line parallel to $y = 2x + 3$ passing through $(1, 2)$ .
Parallel lines have the same slope. The slope of the given line is 2.
Use the point-slope form with point $(1, 2)$ and slope 2:
$y - 2 = 2(x - 1)$ $y - 2 = 2x - 2$ $y = 2x$
The equation of the line is: $y = 2x$

STEP 7

1.5 (Free Response): Write the equation of a line passing through the points $(0, 4)$ and $(2, 8)$ .
First, find the slope $m$ :
$m = \frac{8 - 4}{2 - 0} = \frac{4}{2} = 2$
Use the point-slope form with point $(0, 4)$ and slope 2:
$y - 4 = 2(x - 0)$ $y - 4 = 2x$ $y = 2x + 4$
The equation of the line is: $y = 2x + 4$

STEP 8

1.6 (Free Response): Find the slope of a line that passes through $(-3, 7)$ and $(4, -1)$ . Finish in Slope Intercept form!
First, find the slope $m$ :
$m = \frac{-1 - 7}{4 - (-3)} = \frac{-8}{7}$
Use the point-slope form with point $(-3, 7)$ and slope $-\frac{8}{7}$ :
$y - 7 = -\frac{8}{7}(x + 3)$ $y - 7 = -\frac{8}{7}x - \frac{24}{7}$ $y = -\frac{8}{7}x - \frac{24}{7} + 7$ $y = -\frac{8}{7}x + \frac{25}{7}$
The equation of the line is: $y = -\frac{8}{7}x + \frac{25}{7}$

STEP 9

2.1 (Multiple Choice): Determine whether the lines $y = 3x + 2$ and $y = -\frac{1}{3}x - 5$ are parallel, perpendicular, or neither.
Compare the slopes of the two lines: - First line slope: $3$ - Second line slope: $-\frac{1}{3}$
Two lines are perpendicular if the product of their slopes is $-1$ :
$3 \times -\frac{1}{3} = -1$
The lines are perpendicular.
The correct answer is: b) Perpendicular

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Studdy Solution

STEP 1

1. We are dealing with linear equations in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We need to understand how to manipulate equations to find slope-intercept form and determine relationships between lines.

STEP 2

STEP 3

STEP 4

1.2 (Multiple Choice): What is the slope-intercept form of a line with slope -5 and y-intercept $(0, 4)$ ?
The slope-intercept form is $y = mx + b$ .
Given: - Slope $m = -5$ - y-intercept $b = 4$
Substitute into the slope-intercept form:
$y = -5x + 4$
The correct answer is: a) $y = -5x + 4$

STEP 5

1.3 (Multiple Choice): Convert the equation $3x - 4y = 12$ to slope-intercept form.
Start with the given equation:
$3x - 4y = 12$
Solve for $y$ :
$-4y = -3x + 12$ $y = \frac{3}{4}x - 3$
The correct answer is: b) $y = \frac{3}{4}x - 3$

STEP 6

STEP 7

STEP 8

STEP 9

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Studdy Solution

STEP 1

1. We are dealing with linear equations in the form y=mx+b y = mx + b y=mx+b, where m m m is the slope and b b b is the y-intercept.2. We need to understand how to manipulate equations to find slope-intercept form and determine relationships between lines.

STEP 2

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

STEP 8

STEP 9

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are dealing with linear equations in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We need to understand how to manipulate equations to find slope-intercept form and determine relationships between lines.