Question

write the slope, y-intercept, and equation for the following
1
slope: ______
y-intercept: ______
equation: ______
2
slope: ______
y-intercept: ______
equation: ______
graph the following linear equations
slope - intercept form: ( y = mx + b )

1. ( y = 4x - 2 )

( m = ) ____ ( b = ) ____

1. ( y = -\frac{2}{3}x + 3 )

( m = ) ____ ( b = ) ____

Explanation:

Problem 3: Analyze \( y = 4x - 2 \)

Step 1: Identify slope (\( m \))

In slope - intercept form \( y=mx + b \), the coefficient of \( x \) is the slope. For \( y = 4x-2 \), comparing with \( y=mx + b \), we have \( m = 4 \).

Step 2: Identify y - intercept (\( b \))

In the equation \( y=mx + b \), the constant term is the y - intercept. For \( y = 4x-2 \), we can rewrite it as \( y=4x+(- 2) \), so \( b=-2 \).

Step 3: Graphing the line

• Plot the y - intercept: The y - intercept is \( b=-2 \), so we plot the point \( (0,-2) \) on the y - axis.
• Use the slope to find another point: The slope \( m = 4=\frac{4}{1} \). The slope is \( \frac{\text{rise}}{\text{run}} \), which means from the point \( (0,-2) \), we move up 4 units (because the numerator of the slope is 4) and then move 1 unit to the right (because the denominator of the slope is 1). This gives us the point \( (0 + 1,-2 + 4)=(1,2) \).
• Draw the line: Connect the points \( (0,-2) \) and \( (1,2) \) (and extend the line in both directions) to graph the line \( y = 4x-2 \).

Step 1: Identify slope (\( m \))

In the slope - intercept form \( y = mx + b \), for the equation \( y=-\frac{2}{3}x + 3 \), the coefficient of \( x \) is the slope. So \( m=-\frac{2}{3} \).

Step 2: Identify y - intercept (\( b \))

In the equation \( y=mx + b \), the constant term is the y - intercept. For \( y=-\frac{2}{3}x + 3 \), \( b = 3 \).

Step 3: Graphing the line

• Plot the y - intercept: The y - intercept is \( b = 3 \), so we plot the point \( (0,3) \) on the y - axis.
• Use the slope to find another point: The slope \( m=-\frac{2}{3}=\frac{- 2}{3} \). The slope is \( \frac{\text{rise}}{\text{run}} \). From the point \( (0,3) \), we move down 2 units (because the numerator of the slope is - 2) and then move 3 units to the right (because the denominator of the slope is 3). This gives us the point \( (0 + 3,3-2)=(3,1) \). We can also move up 2 units and left 3 units from \( (0,3) \) to get another point \( (0 - 3,3 + 2)=(-3,5) \).
• Draw the line: Connect the points (e.g., \( (0,3) \) and \( (3,1) \)) and extend the line in both directions to graph the line \( y=-\frac{2}{3}x + 3 \).

Answer:

\( m=\boldsymbol{4} \), \( b=\boldsymbol{-2} \) (For graphing, plot \( (0, - 2) \) and use slope \( 4 \) to find other points and draw the line)

Problem 4: Analyze \( y=-\frac{2}{3}x + 3 \) (assuming the equation is \( y =-\frac{2}{3}x+3 \) from the image)