Lines - SSAT Upper Level Quantitative

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Question

Give the equation of the line through and .

Answer

First, find the slope:

Apply the point-slope formula:

Rewriting in standard form:

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Question

A line can be represented by . What is the slope of the line that is perpendicular to it?

Answer

You will first solve for Y, to get the equation in form.

represents the slope of the line, which would be .

A perpendicular line's slope would be the negative reciprocal of that value, which is .

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Question

Lines

Examine the above diagram. What is ?

Answer

Use the properties of angle addition:

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Question

Give the equation of a line that passes through the point and has an undefined slope.

Answer

A line with an undefined slope has equation for some number ; since this line passes through a point with -coordinate 4, then this line must have equation

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Question

Give the equation of a line that passes through the point and has slope 1.

Answer

We can use the point slope form of a line, substituting .

or

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Question

Find the equation the line goes through the points and .

Answer

First, find the slope of the line.

Now, because the problem tells us that the line goes through , our y-intercept must be .

Putting the pieces together, we get the following equation:

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Question

A line passes through the points and . Find the equation of this line.

Answer

To find the equation of a line, we need to first find the slope.

Now, our equation for the line looks like the following:

To find the y-intercept, plug in one of the given points and solve for . Using , we get the following equation:

Solve for .

Now, plug the value for into the equation.

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Question

What is the equation of a line that passes through the points and ?

Answer

First, we need to find the slope of the line.

Next, find the -intercept. To find the -intercept, plug in the values of one point into the equation , where is the slope that we just found and is the -intercept.

Solve for .

Now, put the slope and -intercept together to get

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Question

Are the following two equations parallel?

Answer

When two lines are parallal, they must have the same slope.

Look at the equations when they are in slope-intercept form, where b represents the slope.

We must first reduce the second equation since all of the constants are divisible by .

This leaves us with . Since both equations have a slope of , they are parallel.

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Question

Reduce the following expression:

Answer

For this expression, you must take each variable and deal with them separately.

First divide you two constants .

Then you move onto and when you divide like exponents you must subtract the exponents leaving you with .

is left by itself since it is already in a natural position.

Whenever you have a negative exponential term, you must it in the denominator.

This leaves the expression of .

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Question

For the line

Which one of these coordinates can be found on the line?

Answer

To test the coordinates, plug the x-coordinate into the line equation and solve for y.

y = 1/3x -7

Test (3,-6)

y = 1/3(3) – 7 = 1 – 7 = -6 YES!

Test (3,7)

y = 1/3(3) – 7 = 1 – 7 = -6 NO

Test (6,-12)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (6,5)

y = 1/3(6) – 7 = 2 – 7 = -5 NO

Test (9,5)

y = 1/3(9) – 7 = 3 – 7 = -4 NO

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Question

Consider the lines described by the following two equations:

4y = 3x2

3y = 4x2

Find the vertical distance between the two lines at the points where x = 6.

Answer

Since the vertical coordinates of each point are given by y, solve each equation for y and plug in 6 for x, as follows:

Taking the difference of the resulting y -values give the vertical distance between the points (6,27) and (6,48), which is 21.

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Question

Solve the following system of equations:

–2x + 3y = 10

2x + 5y = 6

Answer

Since we have –2x and +2x in the equations, it makes sense to add the equations together to give 8y = 16 yielding y = 2. Then we substitute y = 2 into one of the original equations to get x = –2. So the solution to the system of equations is (–2, 2)

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Question

Which of the following sets of coordinates are on the line y=3x-4?

Answer

(2,2) when plugged in for y and x make the linear equation true, therefore those coordinates fall on that line.

y=3x-4

Because this equation is true, the point must lie on the line. The other given answer choices do not result in true equalities.

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Question

Which of the following points can be found on the line \small y=3x+2?

Answer

We are looking for an ordered pair that makes the given equation true. To solve, plug in the various answer choices to find the true equality.

Because this equality is true, we can conclude that the point lies on this line. None of the other given answer options will result in a true equality.

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Question

Which of the following points is on both the line

and the line

Answer

In the multiple choice format, you can just plug in these points to see which satisfies both equations. and work for the first but not the second, and and work for the second but not the first. Only works for both.

Alternatively (or if you were in a non-multiple choice scenario), you could set the equations equal to each other and solve for one of the variables. So, for instance,

and

so

Now you can solve and get . Plug this back into either of the original equations and get .

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Question

A line has the equation . Which of the following points lies on the line?

Answer

Plug the x-coordinate of an answer choice into the equation to see if the y-coordinate matches with what comes out of the equation.

For ,

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Question

Which of the following points lies on the line with equation ?

Answer

To find which point lies on the line, plug in the x-coordinate value of an answer choice into the equation. If the y-coordinate value that comes out of the equation matches that of the answer choice, then the point is on the line.

For ,

So then, lies on the line.

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Question

Which of the following points is on the line with the equation ?

Answer

To find if a point is on the line, plug in the x-coordinate of the answer choice into the given equation. If the resulting value for the y-coordinate matches that of the answer choice, then that point is on the line.

For ,

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Question

Which of the following points lies on the line with the equation ?

Answer

To find if a point is on the line, plug in the x-coordinate of the answer choice into the given equation. If the resulting value for the y-coordinate matches that of the answer choice, then that point is on the line.

For ,

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