Question

equations of a line quiz \\(\alpha\\)

slope intercept form: \\(y = mx + b\\)

slope point form: \\(y - y_1 = m(x - x_1)\\)

general form: \\(ax + by + c = 0\\)

1. what is the equation of a line with y intercept of 6 and a slope of \\(\frac{-5}{2}\\). graph the equation.

1. what is the equation for this graph?

Explanation:

⚡ Using what you learned: slope-intercept and standard form

Step 1: Write the equation for Question 1

Using the slope-intercept form \( y = mx + b \), where the slope \( m = \frac{-5}{2} \) and the y-intercept \( b = 6 \):

$$ y = -\frac{5}{2}x + 6 $$

Step 2: Identify key points to graph Question 1

• y-intercept: \( (0, 6) \)
• Using the slope \( \frac{\text{rise}}{\text{run}} = \frac{-5}{2} \):

From \( (0, 6) \), move down \( 5 \) units and right \( 2 \) units to find a second point:

$$ (0 + 2, 6 - 5) = (2, 1) $$

• Another point:

From \( (2, 1) \), move down \( 5 \) units and right \( 2 \) units:

$$ (2 + 2, 1 - 5) = (4, -4) $$

Plot the points \( (0, 6) \), \( (2, 1) \), and \( (4, -4) \) on the grid and draw a straight line through them.

Step 3: Find the equation for Question 2

• Identify the y-intercept (\(b\)):

Looking at the graph, the line crosses the y-axis at \( (0, 2) \). Thus, \( b = 2 \).

• Identify another clear point on the grid:

The line passes perfectly through the grid intersection at \( (3, 3) \), \( (6, 4) \), and \( (9, 5) \).

• Calculate the slope (\(m\)):

Using the points \( (0, 2) \) and \( (3, 3) \):

$$ m = \frac{3 - 2}{3 - 0} = \frac{1}{3} $$

• Write the equation:

$$ y = \frac{1}{3}x + 2 $$

Answer:

Question 1:

• Equation: \( y = -\frac{5}{2}x + 6 \)
• Graphing instructions: Plot the y-intercept at \( (0, 6) \). From there, use the slope to go down \( 5 \) units and right \( 2 \) units to plot a second point at \( (2, 1) \). Draw a straight line through these points.

Question 2:

• Equation: \( y = \frac{1}{3}x + 2 \)