Instructions: Click on each question to see the options and after solving you can click to see the explanations
Slope-Intercept Form
1. What is the slope and y-intercept of the line given by the equation \( y = 2x + 3 \)?
A. Slope: 2, Y-intercept: 3
B. Slope: 3, Y-intercept: 2
C. Slope: 1, Y-intercept: 3
D. Slope: 2, Y-intercept: 0
The equation is in the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, the slope is \( 2 \) and the y-intercept is \( 3 \).
2. Write the equation of a line in slope-intercept form that has a slope of -1 and passes through the point (2, 4).
A. \( y = -x + 6 \)
B. \( y = -x + 4 \)
C. \( y = -x + 2 \)
D. \( y = x + 4 \)
Using the point-slope formula \( y - y_1 = m(x - x_1) \): \[ y - 4 = -1(x - 2) \] simplifies to \( y = -x + 6 \).
3. Determine the y-intercept of the line represented by the equation \( 4x - 2y = 8 \).
A. 4
B. -4
C. 2
D. 0
To find the y-intercept, set \( x = 0 \): \[ 4(0) - 2y = 8 \] gives \( y = -4 \).
4. If the slope of a line is 3 and it passes through the point (1, 2), what is the equation in slope-intercept form?
A. \( y = 3x + 1 \)
B. \( y = 3x - 1 \)
C. \( y = 3x + 2 \)
D. \( y = 3x + 5 \)
Using the point-slope form: \[ y - 2 = 3(x - 1) \] gives \( y = 3x - 1 \).
5. Which of the following equations represents a line parallel to \( y = -2x + 5 \)?
A. \( y = -2x + 1 \)
B. \( y = 2x + 5 \)
C. \( y = -3x + 5 \)
D. \( y = 0.5x + 5 \)
Lines that are parallel have the same slope. The slope of the line \( y = -2x + 5 \) is -2, so a parallel line would also have a slope of -2.
6. Given the line \( y = 4x - 8 \), what is the slope of the line perpendicular to it?
A. -0.25
B. 4
C. 0.25
D. -4
The slope of a line perpendicular to another is the negative reciprocal. The slope of \( y = 4x - 8 \) is 4, so the perpendicular slope is \( -\frac{1}{4} = -0.25 \).
Point-Slope Form
8. Convert the point-slope form equation \( y - 5 = -3(x - 2) \) to slope-intercept form.
A. \( y = -3x + 11 \)
B. \( y = -3x + 11 \)
C. \( y = 3x - 1 \)
D. \( y = 3x + 11 \)
Distributing gives: \( y - 5 = -3x + 6 \) then adding 5 gives: \( y = -3x + 11 \).
9. Find the slope of the line represented by the point-slope form equation \( y - 1 = 4(x + 2) \).
A. 4
B. -4
C. 1
D. 0
The slope is the coefficient of \( x \) in the point-slope form equation. Here, the slope is \( 4 \).
10. Given the point-slope form equation \( y - 2 = -5(x + 1) \), what is the y-intercept?
A. 2
B. -3
C. -5
D. 0
Convert to slope-intercept form: \( y = -5x - 5 - 2 \) gives \( y = -5x - 7 \), so the y-intercept is -7.
11. A line has the equation \( y - 4 = 3(x - 2) \). What is the equation in standard form?
A. \( 3x + y = 10 \)
B. \( 3x - y = -10 \)
C. \( 3x - y = -2 \)
D. \( x + 3y = 10 \)
Distributing gives \( y - 4 = 3x - 6 \) or \( 3x - y = 4 \). Rearranging yields the standard form \( 3x - y = -2 \).
12. If a line has the point (3, 4) and slope of -2, what is the point-slope form of the equation?
A. \( y - 4 = -2(x - 3) \)
B. \( y - 3 = -2(x - 4) \)
C. \( y - 2 = -3(x - 3) \)
D. \( y - 4 = 2(x + 3) \)
Using the point-slope formula \( y - y_1 = m(x - x_1) \), we get \( y - 4 = -2(x - 3) \).
Standard Form of Linear Equations
13. Convert the equation \( 2x + 3y = 6 \) to slope-intercept form.
A. \( y = -\frac{2}{3}x + 2 \)
B. \( y = \frac{2}{3}x + 2 \)
C. \( y = \frac{3}{2}x - 6 \)
D. \( y = -3x + 6 \)
Rearranging gives: \( 3y = -2x + 6 \) or \( y = -\frac{2}{3}x + 2 \).
14. Determine the standard form of the equation of a line with slope 5 that passes through the point (2, 3).
A. \( 5x + y = 23 \)
B. \( 5x - y = 7 \)
C. \( 2x - 5y = -13 \)
D. \( x + 5y = 15 \)
Using point-slope form: \( y - 3 = 5(x - 2) \), simplifying gives \( 5x - y = 7 \).
15. Which of the following is the standard form of the equation for a line that has a y-intercept of -2 and a slope of 1?
A. \( x - y = 2 \)
B. \( y + 2 = x \)
C. \( 2y - x = -4 \)
D. \( y + 2 = -x \)
The slope-intercept form is \( y = x - 2 \) or \( x - y = 2 \).
16. Find the x-intercept of the equation \( 3x - 4y = 12 \).
A. 4
B. -4
C. 3
D. -3
To find the x-intercept, set \( y = 0 \): \( 3x = 12 \) gives \( x = 4 \).
17. What is the equation in standard form for a line with a slope of -1 and passing through the point (2, 1)?
A. \( -x + y = 1 \)
B. \( x + y = 3 \)
C. \( x + y = 3 \)
D. \( x - y = -1 \)
Using point-slope form: \( y - 1 = -1(x - 2) \) gives \( y = -x + 3 \) or rearranged to \( x + y = 3 \).
Graphing Linear Equations
18. Graph the equation \( y = 2x - 4 \). What is the slope and y-intercept of this line?
A. Slope: 2, Y-intercept: -4
B. Slope: -2, Y-intercept: 4
C. Slope: 4, Y-intercept: -2
D. Slope: 2, Y-intercept: 4
In the equation \( y = 2x - 4 \), the slope is \( 2 \) and the y-intercept is \( -4 \).
19. What are the coordinates of the x-intercept of the line \( 3x - 2y = 6 \)?
A. (2, 0)
B. (0, -3)
C. (0, 2)
D. (-2, 0)
Set \( y = 0 \): \( 3x = 6 \) gives \( x = 2 \), so the x-intercept is \( (2, 0) \).
20. What is the slope of the line represented by the graph below?
A. 1/2
B. -1
C. 0
D. 2
The slope is the ratio of the rise over the run. If the graph shows a line that rises 1 unit for every 2 units it runs, the slope is 1. Slope = change in y / change in x
21. A line passes through the points (1, 2) and (3, 6). What is the equation of the line in slope-intercept form?
A. \( y = 2x \)
B. \( y = 2x - 1 \)
C. \( y = 3x - 1 \)
D. \( y = x + 2 \)
The slope between points (1, 2) and (3, 6) is \( \frac{6-2}{3-1} = 2 \). The y-intercept can be found by substituting \( x = 1 \): \( y = 2(1) = 2 \). Thus, the equation is \( y = 2x \).
22. Given the line graph, what is the y-coordinate of the point where the line crosses the y-axis?
A. 4
B. -2
C. 0
D. -1
The y-coordinate where the line crosses the y-axis is the y-intercept. Check the graph to find this value.
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