How to find the equation of a line - SAT Math

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Question

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 ?

Answer

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.

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Question

Solve the equation for x and y.

x/y = 30

x + y = 5

Answer

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,

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Question

Solve the equation for x and y.

x – y = 26/17

2_x_ + 3_y_ = 2

Answer

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.

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Question

What is the equation for a line with endpoints (-1, 4) and (2, -5)?

Answer

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

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Question

Given the graph of the line below, find the equation of the line.

Answer

To solve this question, you could use two points such as (1.2,0) and (0,-4) to calculate the slope which is 10/3 and then read the y-intercept off the graph, which is -4.

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Question

Find the equation of the line shown in the graph below:

Answer

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.

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Question

Solve the equation for x and y.

–x – 4_y_ = 245

5_x_ + 2_y_ = 150

Answer

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.

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Question

Solve the equation for x and y.

y + 5_x_ = 40

x – y = –10

Answer

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.

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Question

Solve the equation for x and y.

xy=30

x – y = –1

Answer

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.

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Question

Let y = 3_x_ – 6.

At what point does the line above intersect the following:

$2x =\frac{2}{3}y+4$

Answer

If we rearrange the second equation it is the same as the first equation. They are the same line.

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Question

A line is defined by the following equation:

What is the slope of that line?

Answer

The equation of a line is

y=mx + b where m is the slope

Rearrange the equation to match this:

7x + 28y = 84

28y = -7x + 84

y = -(7/28)x + 84/28

y = -(1/4)x + 3

m = -1/4

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Question

If the coordinates (3, 14) and (_–_5, 15) are on the same line, what is the equation of the line?

Answer

First solve for the slope of the line, m using y=mx+b

m = (y2 – y1) / (x2 – x1)

= (15 – 14) / (_–_5 _–_3)

= (1 )/( _–_8)

=_–_1/8

y = –(1/8)x + b

Now, choose one of the coordinates and solve for b:

14 = –(1/8)3 + b

14 = _–_3/8 + b

b = 14 + (3/8)

b = 14.375

y = –(1/8)x + 14.375

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Question

What is the equation of a line that goes through (4, 1) and (–2, –2)?

Answer

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

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Question

What line goes through the points (1, 1) and (–2, 3)?

Answer

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.

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Question

Which line passes through the points (0, 6) and (4, 0)?

Answer

P1 (0, 6) and P2 (4, 0)

First, calculate the slope: m = rise ÷ run = (y2 – y1)/(x2 – x1), so m = –3/2

Second, plug the slope and one point into the slope-intercept formula:

y = mx + b, so 0 = –3/2(4) + b and b = 6

Thus, y = –3/2x + 6

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Question

If angle A is 1/3 the size of angle B, then what is angle A?

Answer

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

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Question

What line goes through the points (1, 3) and (3, 6)?

Answer

If P1(1, 3) and P2(3, 6), then calculate the slope by m = rise/run = (y2 – y1)/(x2 – x1) = 3/2

Use the slope and one point to calculate the intercept using y = mx + b

Then convert the slope-intercept form into standard form.

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Question

What is the equation of a line that passes through coordinates $\dpi{100} \small (2,6)$ and $\dpi{100} \small (3,5)$ ?

Answer

Our first step will be to determing the slope of the line that connects the given points.

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-5}{2-3}=\frac{1}{-1}=-1$

Our slope will be . Using slope-intercept form, our equation will be . Use one of the give points in this equation to solve for the y-intercept. We will use $\dpi{100} \small (2,6)$ .

Now that we know the y-intercept, we can plug it back into the slope-intercept formula with the slope that we found earlier.

This is our final answer.

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Question

What is the equation of the line that passes through the points (4,7) and (8,10)?

Answer

In order to find the equation of the line, we will first need to find the slope between the two points through which it passes. The slope, , of a line that passes through the points and is given by the formula below:

$\text{slope}=m=\frac{y_2-y_1}{x_2-x_1}$

We are given our two points, (4,7) and (8,10), allowing us to calculate the slope.

$m=\frac{10-7}{8-4}=\frac{3}{4}$

Next, we can use point slope form to find the equation of a line with this slope that passes through one of the given points. We will use (4,7).

$y-7=\frac{3}{4}(x-4)$

Multiply both sides by four to eliminate the fraction, and simplify by distribution.

$4(y-7)=4(\frac{3}{4})(x-4)$

Subtract from both sides and add twelve to both sides.

This gives our final answer:

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Question

What is the slope-intercept form of $\dpi{100} \small 8x-2y-12=0$ ?

Answer

The slope intercept form states that $\dpi{100} \small y=mx+b$ . In order to convert the equation to the slope intercept form, isolate $\dpi{100} \small y$ on the left side:

$\dpi{100} \small 8x-2y=12$

$\dpi{100} \small -2y=-8x+12$

$\dpi{100} \small y=4x-6$

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