Math Snap

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:
$$y = -\frac{8}{7}x + \frac{25}{7}$$

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