Let's go through each question one by one:
4. Find the rate of change
Two pounds of apples cost $3.98, so the price per pound is: \[ \text{Price per pound} = \frac{3.98}{2} = 1.99 \text{ per pound} \] Six pounds costing $11.94 confirms this: \[ \text{Price per pound} = \frac{11.94}{6} = 1.99 \text{ per pound} \] Thus, the correct answer is 1.99/lb.
5. Tell whether y varies directly with x, and write an equation for the direct variation
Direct variation means \( y = kx \), where \( k \) is a constant. Without specific values for \( x \) and \( y \), we can’t determine the equation precisely. But since the option gives \( y = 1.25x \), we can suggest; therefore:
- Yes, if we determine \( k = 1.25 \), then y = 1.25x.
6. Find the slope and y-intercept for the equation \( y = -6.75x + 8.54 \)
The equation is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
- Slope = -6.75
- y-intercept = 8.54 Thus, the answer is Slope -6.75, y - int = 8.54.
7. Write an equation for M = 4 and b = -2
Using the slope-intercept form \( y = mx + b \): \[ y = 4x - 2 \] So, the answer is y = 4x - 2.
8. Write an equation in slope-intercept form that passes through (10, 2) and (-2, -2)
First, calculate the slope (m) using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \): \[ m = \frac{-2 - 2}{-2 - 10} = \frac{-4}{-12} = \frac{1}{3} \] Using point-slope form \( y - y_1 = m(x - x_1) \) using the point (10,2): \[ y - 2 = \frac{1}{3}(x - 10) \implies y = \frac{1}{3}x - \frac{10}{3} + 2 \implies y = \frac{1}{3}x - \frac{10}{3} + \frac{6}{3} = \frac{1}{3}x - \frac{4}{3} \] Thus, the answer is y = 1/3 x - 4/3.
9. Find the x and y-intercepts of the equation \( 2x + 3y = -6 \)
- For the x-intercept, set \( y = 0 \): \[ 2x + 3(0) = -6 \implies 2x = -6 \implies x = -3 \quad \text{(Point: -3, 0)} \]
- For the y-intercept, set \( x = 0 \): \[ 2(0) + 3y = -6 \implies 3y = -6 \implies y = -2 \quad \text{(Point: 0, -2)} \] Thus, the answer is -3, -2.
10. Is the graph of the equation \( x = 0 \) a horizontal or vertical line?
The equation \( x = 0 \) is vertical because it represents all points where x is 0. So, the answer is vertical.
11. Is the graph of the equation \( y = -0.25 \) a horizontal or vertical line?
The equation \( y = -0.25 \) is horizontal because it represents all points where y is -0.25. So, the answer is horizontal.
12. Write the equation in standard form from \( y - 4 = 5(x - 8) \)
Start with the given equation: \[ y - 4 = 5(x - 8) \implies y - 4 = 5x - 40 \implies y = 5x - 36 \] Now, rearranging to standard form \( Ax + By = C \): \[ -5x + y = -36 \quad \text{(or multiplying by -1 gives: } 5x - y = 36\text{)} \] The final answer is 5x - y = 36.
13. Write the equation of the line that passes through (-8,6) and is parallel to \( y = -\frac{1}{4}x + 5 \)
Since it is parallel, it has the same slope of \( -\frac{1}{4} \): Using point-slope form: \[ y - 6 = -\frac{1}{4}(x + 8) \implies y - 6 = -\frac{1}{4}x - 2 \implies y = -\frac{1}{4}x + 4 \] Thus, the answer is y = -\frac{1}{4}x + 4.
14. Determine if the lines are parallel, perpendicular, or neither for:
- \( y = -6x - 8 \) (slope -6)
- \( -x + 6y = 12 \) (convert to slope-intercept): \[ 6y = x + 12 \implies y = \frac{1}{6}x + 2 \quad \text{(slope = }\frac{1}{6}) \] The product of the slopes \( (-6) \times \frac{1}{6} = -1 \) indicates they are perpendicular.
15. Determine if the lines are parallel, perpendicular, or neither for:
- \( y = 4x + 12 \) (slope 4)
- \( x + 4y = 32 \) (convert to slope-intercept): \[ 4y = -x + 32 \implies y = -\frac{1}{4}x + 8 \quad \text{(slope = } -\frac{1}{4}) \] The product of the slopes \( 4 \times (-\frac{1}{4}) = -1 \) indicates they are perpendicular.
I hope this helps! If you have any further questions, feel free to ask!