slope
Why is the slope undefined for a vertical line?
The slope is the ratio of the vertical change to the horizontal change. In a vertical line, the horizontal change is zero, resulting in a division by zero, which is undefined.
For the equation �=2�+3y=2x+3, what is the y-intercept?
The y-intercept is the value of �y when �=0x=0. Substituting �=0x=0 into the equation: �=2(0)+3=3y=2(0)+3=3 So, the y-intercept is 3.
Solve the equation: 3�−7=143x−7=14.
3x−7=14 3�=213x=21 �=7x=7
Solve the system of equations: 2�+�=82x+y=8 3�−2�=13x−2y=1
Multiply the first equation by 2 and add it to the second equation to eliminate �y:
4�+2�+3�−2�=16+14x+2y+3x−2y=16+1
7�=177x=17
�=177x=717
Graph the linear equation �=2�−3y=2x−3. Identify the slope and y-intercept.
The graph is a straight line with a slope of 2 and a y-intercept of -3.
If the equation of a line is �=−3�+7y=−3x+7, what is the slope of the line?
The slope (�m) is the coefficient of the �x term, so in this case, �=−3m=−3.
If a line passes through the point (0, -5), what is the y-intercept of the line?
The point (0, -5) means the y-intercept is -5.
Factor the quadratic equation: �2−5�+6=0x2−5x+6=0.
(x−2)(x−3)=0 �=2x=2 or �=3x=3
Solve the system of equations: 4�−3�=54x−3y=5 2�+�=72x+y=7
Multiply the second equation by 3 and add it to the first equation to eliminate �y:
4�−3�+6�+3�=5+214x−3y+6x+3y=5+21
10�=2610x=26
�=135x=513
Sketch the graph of the quadratic function �=�2−4�+4y=x2−4x+4. Identify the vertex and axis of symmetry.
The graph is a parabola with vertex at (2, 0) and the axis of symmetry is �=2x=2.
If two lines are perpendicular, and one line has a slope of 2, what is the slope of the other line?
For perpendicular lines, the product of their slopes is -1. Therefore, if one line has a slope of 2, the other line would have a slope of −12−21.
Given the equation 3�−2�=123y−2x=12, find the y-intercept.
To find the y-intercept, set �=0x=0 and solve for �y: 3�−2(0)=123y−2(0)=12 3�=123y=12 �=4y=4 So, the y-intercept is 4.
Simplify the expression: 2(4�−6)+3�2(4x−6)+3x.
2(4x−6)+3x=8x−12+3x=11x−12
Solve the system of equations: 3�+2�=103x+2y=10 2�−�=32x−y=3
Multiply the second equation by 2 and add it to the first equation to eliminate �y: 3�+2�+4�−2�=10+63x+2y+4x−2y=10+6 7�=167x=16 �=167x=716
Substitute �x back into the second equation: 2(167)−�=32(716)−y=3 �=57y=75
So, the solution is �=167x=716 and �=57y=75.
Graph the absolute value function �=∣�−2∣y=∣x−2∣. Identify the vertex and the behavior of the graph for �<2x<2 and �>2x>2.
The graph is a V-shaped graph with the vertex at (2, 0). For �<2x<2, the graph is decreasing, and for �>2x>2, the graph is increasing.
If two lines are perpendicular, and one line has a slope of 2, what is the slope of the other line?
For perpendicular lines, the product of their slopes is -1. Therefore, if one line has a slope of 2, the other line would have a slope of −12−21.
If a line is represented by the equation �=−0.5�+7y=−0.5x+7, what is the y-intercept?
The y-intercept is the constant term in the equation. Therefore, the y-intercept is 7.
Write the equation of a line in slope-intercept form given the slope �=−2m=−2 and the y-intercept �=5b=5.
y=−2x+5
Solve the system of equations: �+2�=3x+2y=3 2�−�=52x−y=5
Multiply the first equation by 2 and add it to the second equation to eliminate �y: 2�+4�+2�−�=6+52x+4y+2x−y=6+5 4�+3�=114x+3y=11
This system has no solution.
Draw the graph of the system of linear equations: 2�+�=52x+y=5 3�−2�=43x−2y=4
The lines intersect at the point (2, 1), which is the solution to the system.
Given the equation 2�+4�=102y+4x=10, rewrite it in slope-intercept form (�=��+�y=mx+b).
2y+4x=10 2�=−4�+102y=−4x+10 �=−2�+5y=−2x+5
For the line with equation 2�+4�=102y+4x=10, find the y-intercept.
To find the y-intercept, set �=0x=0 and solve for �y: 2�+4(0)=102y+4(0)=10 2�=102y=10 �=5y=5 So, the y-intercept is 5.
Solve the system of equations: 2�+�=102x+y=10 3�−2�=43x−2y=4
Multiply the first equation by 2 and add it to the second equation to eliminate �y: 4�+2�+3�−2�=20+44x+2y+3x−2y=20+4 7�=247x=24 �=247x=724
Substitute �x back into the first equation: 2(247)+�=102(724)+y=10 �=27y=72
So, the solution is �=247x=724 and �=27y=72.
Solve the system of equations: 2�−�=42x−y=4 4�+2�=104x+2y=10
Divide the second equation by 2 to make the coefficients of �y the same, then subtract the first equation from the second to eliminate �y: 2�−�=42x−y=4 2�+�=52x+y=5 (2�+�)−(2�−�)=5−4(2x+y)−(2x−y)=5−4 2�=12y=1 �=12y=21
Substitute �y back into the first equation: 2�−12=42x−21=4 �=94x=49
So, the solution is �=94x=49 and �=12y=21.
Graph the inequality �>−2�+3y>−2x+3. Identify the shaded region.
The graph is a dashed line with a slope of -2 and a y-intercept of 3. The shaded region is above the line.