Slope and Line Equations - PSAT Math
Card 0 of 189
Solve the equation for x and y.
xy=30
x – y = –1
Solve the equation for x and y.
xy=30
x – y = –1
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Again the same process is required. This problem however involved multiplying x by y so is a bit different. We end up with two possible solutions. Derive y=x+1 and solve in the same manner as the ones above. The graph below illustrates the solution.
Again the same process is required. This problem however involved multiplying x by y so is a bit different. We end up with two possible solutions. Derive y=x+1 and solve in the same manner as the ones above. The graph below illustrates the solution.
Solve the equation for x and y.
x/y = 30
x + y = 5
Solve the equation for x and y.
x/y = 30
x + y = 5
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Similar problem to the one before, with x being divided by y instead of multiplied. Solve in the same manner but keep in mind the way that x/y is graphed. We end up solving for one solution. The graph below illustrates the solution,
Similar problem to the one before, with x being divided by y instead of multiplied. Solve in the same manner but keep in mind the way that x/y is graphed. We end up solving for one solution. The graph below illustrates the solution,
Solve the equation for x and y.
x – y = 26/17
2_x_ + 3_y_ = 2
Solve the equation for x and y.
x – y = 26/17
2_x_ + 3_y_ = 2
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Straightforward problem that presents two unknowns with two equations. The student will need to deal with the fractions correctly to get this one right. Other than the fraction the problem is solved in the exact same manner as the rest in this set. The graph below illustrates the solution.
Straightforward problem that presents two unknowns with two equations. The student will need to deal with the fractions correctly to get this one right. Other than the fraction the problem is solved in the exact same manner as the rest in this set. The graph below illustrates the solution.
What is the equation for a line with endpoints (-1, 4) and (2, -5)?
What is the equation for a line with endpoints (-1, 4) and (2, -5)?
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First we need to find the slope. Slope (m) = (y2 - y1)/(x2 - x1). Substituting in our values (-5 - 4)/(2 - (-1)) = -9/3 = -3 so slope = -3. The formula for a line is y = mx +b. We know m = -3 so now we can pick one of the two points, substitute in the values for x and y, and find b. 4 = (-3)(-1) + b so b = 1. Our formula is thus y = -3x + 1
First we need to find the slope. Slope (m) = (y2 - y1)/(x2 - x1). Substituting in our values (-5 - 4)/(2 - (-1)) = -9/3 = -3 so slope = -3. The formula for a line is y = mx +b. We know m = -3 so now we can pick one of the two points, substitute in the values for x and y, and find b. 4 = (-3)(-1) + b so b = 1. Our formula is thus y = -3x + 1
Solve the equation for x and y.
–x – 4_y_ = 245
5_x_ + 2_y_ = 150
Solve the equation for x and y.
–x – 4_y_ = 245
5_x_ + 2_y_ = 150
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While solving the problem requires the same method as the ones above, this is one is more complicated because of the more complex given equations. Start of by deriving a substitute for one of the unknowns. From the second equation we can derive y=75-(5x/2). Since 2y = 150 -5x, we divide both sides by two and find our substitution for y. Then we enter this into the first equation. We now have –x-4(75-(5x/2))=245. Distribute the 4. So we get –x – 300 + 10x = 245. So 9x =545, and x=545/9. Use this value for x and solve for y. The graph below illustrates the solution.
While solving the problem requires the same method as the ones above, this is one is more complicated because of the more complex given equations. Start of by deriving a substitute for one of the unknowns. From the second equation we can derive y=75-(5x/2). Since 2y = 150 -5x, we divide both sides by two and find our substitution for y. Then we enter this into the first equation. We now have –x-4(75-(5x/2))=245. Distribute the 4. So we get –x – 300 + 10x = 245. So 9x =545, and x=545/9. Use this value for x and solve for y. The graph below illustrates the solution.
Solve the equation for x and y.
y + 5_x_ = 40
x – y = –10
Solve the equation for x and y.
y + 5_x_ = 40
x – y = –10
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This one is a basic problem with two unknowns in two equations. Derive y=x+10 from the second equation and replace the y in first equation to solve the problem. So, x+10+5x=40 and x = 5. X-y= -10 so y=15. The graph below illustrates the solution.
This one is a basic problem with two unknowns in two equations. Derive y=x+10 from the second equation and replace the y in first equation to solve the problem. So, x+10+5x=40 and x = 5. X-y= -10 so y=15. The graph below illustrates the solution.
Based on the table below, when x = 5, y will equal
x y -1 3 0 1 1 -1 2 -3
Based on the table below, when x = 5, y will equal
| x | y |
|---|---|
| -1 | 3 |
| 0 | 1 |
| 1 | -1 |
| 2 | -3 |
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Use 2 points from the chart to find the equation of the line.
Example: (–1, 3) and (1, –1)
Using the formula for the slope, we find the slope to be –2. Putting that into our equation for a line we get y = –2x + b. Plug in one of the points for x and y into this equation in order to find b. b = 1.
The equation then will be: y = –2x + 1.
Plug in 5 for x in order to find y.
y = –2(5) + 1
y = –9
Use 2 points from the chart to find the equation of the line.
Example: (–1, 3) and (1, –1)
Using the formula for the slope, we find the slope to be –2. Putting that into our equation for a line we get y = –2x + b. Plug in one of the points for x and y into this equation in order to find b. b = 1.
The equation then will be: y = –2x + 1.
Plug in 5 for x in order to find y.
y = –2(5) + 1
y = –9
What is the slope of a line that runs through points: (-2, 5) and (1, 7)?
What is the slope of a line that runs through points: (-2, 5) and (1, 7)?
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The slope of a line is defined as a change in the y coordinates over a change in the x coordinates (rise over run).
To calculate the slope of a line, use the following formula: 
The slope of a line is defined as a change in the y coordinates over a change in the x coordinates (rise over run).
To calculate the slope of a line, use the following formula:
A line passes through the points (–3, 5) and (2, 3). What is the slope of this line?
A line passes through the points (–3, 5) and (2, 3). What is the slope of this line?
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The slope of the line that passes these two points are simply ∆y/∆x = (3-5)/(2+3) = -2/5
The slope of the line that passes these two points are simply ∆y/∆x = (3-5)/(2+3) = -2/5
Which of the following lines intersects the y-axis at a thirty degree angle?
Which of the following lines intersects the y-axis at a thirty degree angle?
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What is a possible slope of line y?
What is a possible slope of line y?
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The slope is negative as it starts in quadrant 2 and ends in quadrant 4. Slope is equivlent to the change in y divided by the change in x. The change in y is greater than the change in x, which implies that the slope must be less than –1, leaving –2 as the only possible solution.
The slope is negative as it starts in quadrant 2 and ends in quadrant 4. Slope is equivlent to the change in y divided by the change in x. The change in y is greater than the change in x, which implies that the slope must be less than –1, leaving –2 as the only possible solution.
What is the slope between 
and 
?
What is the slope between
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Let $P_{1}$=(8,3) and $P_{2}$=(5,7)
m = $(y_{2}$ - $y_{1}$) div $(x_{2}$ - $x_{1}$) so the slope becomes 
.
Let $P_{1}$=(8,3) and $P_{2}$=(5,7)
m = $(y_{2}$ - $y_{1}$) div $(x_{2}$ - $x_{1}$) so the slope becomes
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Refer to above red line. What is its slope?
Refer to above red line. What is its slope?
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The slope of a line. given two points 
can be calculated using the slope formula
Set 
:
The slope of a line. given two points
Set
Which of the following equations has as its graph a line with slope 4?
Which of the following equations has as its graph a line with slope 4?
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For each equation, solve for 
and express in the slope-intercept form 
. The coefficient of 
will be the slope.
Slope: 
Slope: 
Slope: 
Slope: 
.
The line of the equation
is the one with slope 4.
For each equation, solve for
Slope:
Slope:
Slope:
Slope:
The line of the equation
is the one with slope 4.
Find the equation of the line shown in the graph below:
Find the equation of the line shown in the graph below:
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Based on the graph the y-intercept is 4. So we can eliminate choice y = x/2 - 4.
The graph is rising to the right which means our slope is positive, so we can eliminate choice y = -1/2x + 4.
Based on the line, if we start at (0,4) and go up 1 then 2 to the right we will be back on the line, meaning we have a slope of (1/2).
Using the slope intercept formula we can plug in y= (1/2)x + 4.
Based on the graph the y-intercept is 4. So we can eliminate choice y = x/2 - 4.
The graph is rising to the right which means our slope is positive, so we can eliminate choice y = -1/2x + 4.
Based on the line, if we start at (0,4) and go up 1 then 2 to the right we will be back on the line, meaning we have a slope of (1/2).
Using the slope intercept formula we can plug in y= (1/2)x + 4.
What is the equation of a line that goes through (4, 1) and (–2, –2)?
What is the equation of a line that goes through (4, 1) and (–2, –2)?
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We will find the equation using slope intercept form: y=mx+b
1. Use the two points to find the slope.
The equation to find the slope of this line using two points is:
Therefore, m = 1/2, so the slope of this line is 1/2.
2. Now that we have the slope, we can use one of the points that were given to find the y intercept. In order to do this, substitute y for the y value of the point, and substitute x for the x value of the point.
Using the point (–2, –2), we now have: –2 = (1/2)(–2) + b.
Simplify the equation to solve for b. b = –1
3. In this line m = 1/2 and b = –1
4. Therefore, y = 1/2x –1
We will find the equation using slope intercept form: y=mx+b
1. Use the two points to find the slope.
The equation to find the slope of this line using two points is:
Therefore, m = 1/2, so the slope of this line is 1/2.
2. Now that we have the slope, we can use one of the points that were given to find the y intercept. In order to do this, substitute y for the y value of the point, and substitute x for the x value of the point.
Using the point (–2, –2), we now have: –2 = (1/2)(–2) + b.
Simplify the equation to solve for b. b = –1
3. In this line m = 1/2 and b = –1
4. Therefore, y = 1/2x –1
What line goes through the points (1, 1) and (–2, 3)?
What line goes through the points (1, 1) and (–2, 3)?
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Let P1 (1, 1) and P2 (–2, 3).
First, find the slope using m = rise ÷ run = (y2 – y1)/(x2 – x1) giving m = –2/3.
Second, substutite the slope and a point into the slope-intercept equation y = mx + b and solve for b giving b = 5/3.
Third, convert the slope-intercept form into the standard form giving 2x + 3y = 5.
Let P1 (1, 1) and P2 (–2, 3).
First, find the slope using m = rise ÷ run = (y2 – y1)/(x2 – x1) giving m = –2/3.
Second, substutite the slope and a point into the slope-intercept equation y = mx + b and solve for b giving b = 5/3.
Third, convert the slope-intercept form into the standard form giving 2x + 3y = 5.
If angle A is 1/3 the size of angle B, then what is angle A?
If angle A is 1/3 the size of angle B, then what is angle A?
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The answer is 22.5.
From the image we can tell that angle a and angle b are complimentary
a + b = 90 and 3a = b
a + 3a = 90
a = 22.5
The answer is 22.5.
From the image we can tell that angle a and angle b are complimentary
a + b = 90 and 3a = b
a + 3a = 90
a = 22.5
What is the equation of the line with a negative slope that passes through the y-intercept and one x-intercept of the graph y = –x_2 – 2_x + 8 ?
What is the equation of the line with a negative slope that passes through the y-intercept and one x-intercept of the graph y = –x_2 – 2_x + 8 ?
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In order to find the equation of the line, we need to find two points on the line. We are told that the line passes through the y-intercept and one x-intercept of y = –x_2 – 2_x + 8.
First, let's find the y-intercept, which occurs where x = 0. We can substitute x = 0 into our equation for y.
y = –(0)2 – 2(0) + 8 = 8
The y-intercept occurs at (0,8).
To determine the x-intercepts, we can set y = 0 and solve for x.
0 = –x_2 – 2_x + 8
–x_2 – 2_x + 8 = 0
Multiply both sides by –1 to minimize the number of negative coefficients.
x_2 + 2_x – 8 = 0
We can factor this by thinking of two numbers that multiply to give us –8 and add to give us 2. Those numbers are 4 and –2.
x_2 + 2_x – 8= (x + 4)(x – 2) = 0
Set each factor equal to zero.
x + 4 = 0
Subtract 4.
x = –4
Now set x – 2 = 0. Add 2 to both sides.
x = 2
The x-intercepts are (–4,0) and (2,0).
However, we don't know which x-intercept the line passes through. But, we are told that the line has a negative slope. This means it must pass through (2,0).
The line passes through (0,8) and (2,0).
We can use slope-intercept form to write the equation of the line. According to slope-intercept form, y = mx + b, where m is the slope, and b is the y-intercept. We already know that b = 8, since the y-intercept is at (0,8). Now, all we need is the slope, which we can find by using the following formula:
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m = (0 – 8)/(2 – 0) = –8/2 = –4
y = mx + b = –4_x_ + 8
The answer is y = –4_x_ + 8.
In order to find the equation of the line, we need to find two points on the line. We are told that the line passes through the y-intercept and one x-intercept of y = –x_2 – 2_x + 8.
First, let's find the y-intercept, which occurs where x = 0. We can substitute x = 0 into our equation for y.
y = –(0)2 – 2(0) + 8 = 8
The y-intercept occurs at (0,8).
To determine the x-intercepts, we can set y = 0 and solve for x.
0 = –x_2 – 2_x + 8
–x_2 – 2_x + 8 = 0
Multiply both sides by –1 to minimize the number of negative coefficients.
x_2 + 2_x – 8 = 0
We can factor this by thinking of two numbers that multiply to give us –8 and add to give us 2. Those numbers are 4 and –2.
x_2 + 2_x – 8= (x + 4)(x – 2) = 0
Set each factor equal to zero.
x + 4 = 0
Subtract 4.
x = –4
Now set x – 2 = 0. Add 2 to both sides.
x = 2
The x-intercepts are (–4,0) and (2,0).
However, we don't know which x-intercept the line passes through. But, we are told that the line has a negative slope. This means it must pass through (2,0).
The line passes through (0,8) and (2,0).
We can use slope-intercept form to write the equation of the line. According to slope-intercept form, y = mx + b, where m is the slope, and b is the y-intercept. We already know that b = 8, since the y-intercept is at (0,8). Now, all we need is the slope, which we can find by using the following formula:
m = (0 – 8)/(2 – 0) = –8/2 = –4
y = mx + b = –4_x_ + 8
The answer is y = –4_x_ + 8.