Question

1. draw a line from each statement to the graph it describes. note the rise and run labeled on each graph.

the line has a slope of –3. the y-intercept is 2. the y-intercept is –3. the line has a slope of 3/4.

1. marcus incorrectly identifies two of the key features of the graph y = 3 – 4x. put an x next to any incorrect statements. correct his errors.

a. the slope of the line is 3.
b. the line goes down from left to right.
c. the y-intercept is –4.
d. to graph the line, plot the y-intercept. then plot another point four units down and one unit right.

1. what is an equation in slope-intercept form for the line that passes through the points (1, –3) and (3, 1)? fill in the missing information.

first, use the two given points to find the slope.
m = (y₂ - y₁)/(x₂ - x₁)
m = (1 - (-3))/(3 - 1) = 4/2 = ______
use the slope and one point to write an equation of the line in slope-intercept form.
y = mx + b \t\t\tslope-intercept form of a linear equation.
____ = ____ + b \tsubstitute (1, –3) for (x₁, y₁) and 2 for m.
b = ______ \t\t\tsolve for b.
an equation in slope-intercept form is ______________

Explanation:

Problem 3 Solution:

Step1: Calculate the slope

We use the slope formula \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Given points \((1, -3)\) and \((3, 1)\), substitute into the formula: \( m=\frac{1 - (-3)}{3 - 1}=\frac{4}{2} = 2 \).

Step2: Substitute into slope - intercept form

The slope - intercept form is \( y=mx + b \). We substitute \( x = 1 \), \( y=-3 \) and \( m = 2 \) into the equation: \( -3=2\times1 + b \).

Step3: Solve for \( b \)

Simplify the equation \( -3 = 2 + b \). Subtract 2 from both sides: \( b=-3 - 2=-5 \).

Step4: Write the equation

Now that we have \( m = 2 \) and \( b=-5 \), the slope - intercept form of the line is \( y = 2x-5 \).

(for Problem 2):

• For statement a: The slope of \( y = 3-4x\) (or \( y=-4x + 3\)) is - 4, not 3. So it's incorrect (X).
• For statement b: Since slope \( m=-4<0 \), the line decreases (goes down left - to - right), so it's correct.
• For statement c: The \( y \)-intercept of \( y=-4x + 3\) is 3, not - 4. So it's incorrect (X).
• For statement d: Slope \( m=-4=\frac{-4}{1} \), so moving 4 units down (rise=-4) and 1 unit right (run = 1) is correct.

(for Problem 1):

• For "The line has a slope of - 3": The right - hand graph has a slope of - 3 (rise=-3, run = 1).
• For "The \( y \)-intercept is 2": The right - hand graph crosses the \( y \)-axis at \( y = 2 \).
• For "The \( y \)-intercept is - 3": The left - hand graph crosses the \( y \)-axis at \( y=-3 \).
• For "The line has a slope of \( \frac{3}{4} \)": The left - hand graph has a slope of \( \frac{3}{4} \) (rise = 3, run = 4).

Answer:

First blank (slope calculation): \( 2 \)

Second part (substitute into \( y=mx + b \)): \( -3=2\times1 + b \) (or \( -3 = 2 + b \))

Third part (solve for \( b \)): \( b=-5 \)

Fourth part (equation): \( y = 2x-5 \)

Problem 2 Solution:

We start by recalling the slope - intercept form of a line \( y=mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. For the line \( y = 3-4x \), we can rewrite it as \( y=-4x + 3 \). So the slope \( m=-4 \) and the \( y \)-intercept \( b = 3 \).

• Statement a: The slope of the line is 3. This is incorrect. The correct slope is - 4. So we put an X next to this statement.
• Statement b: The line goes down from left to right. Since the slope \( m=-4<0 \), the line is decreasing (goes down from left to right). This statement is correct.
• Statement c: The \( y \)-intercept is - 4. This is incorrect. The correct \( y \)-intercept is 3. So we put an X next to this statement.
• Statement d: To graph the line, plot the \( y \)-intercept. Then plot another point four units down and one unit right. Since the slope is \( - 4=\frac{-4}{1} \) (rise over run, where rise=-4 and run = 1), this means we go 4 units down (because rise is negative) and 1 unit to the right. This statement is correct.