x and y Intercept - ACT Math

Card 0 of 19

Question

Find the slope of the following line: 6_x –_ 4_y_ = 10

Answer

Putting the equation in y = mx + b form we obtain y = 1.5_x_ – 2.5.

The slope is 1.5.

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Question

What is the x-intercept of the line in the standard $\left ( x,y \right )$ coordinate plane for the following equation?

$\frac{1}{2}y+10=6x-2$

Answer

This question is asking us to find the x-intercept. Remember that the y-value is equal to zero at the x-intercept. Substitute zero in for the y-variable in the equation and solve for the x-variable.

$\frac{1}{2}(0)+10=6x-2$

Add 2 to both sides of the equation.

Divide both sides of the equation by 6.

$\frac{12}{6}=\frac{6x}{6}$

The line crosses the x-axis at 2.

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Question

What is the equation of a line that has an x-intercept of 4 and a y-intercept of -6?

Answer

The equation of a line can be written in the following form:

In this formula, m is the slope, and b represents the y-intercept. The problem provides the y-intercept; therefore, we know the following information:

We can calculate the slope of the line, because if any two points on the function are known, then the slope can be calculated. Generically, the slope of a line is defined as the function's rise over run, or more technically, the changes in the y-values over the changes in the x-values. It is formally written as the following equation:

$\text{Slope}=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

The problem provides the two intercepts of the line, which can be written as $\left ( 4,0\right )$ and $\left ( 0,-6\right )$ . Substitute these points into the equation for slope and solve:

$\text{Slope}=\frac{(-6) - 0}{0-4}$

$\text{Slope}=\frac{-6}{-4}$

$\text{Slope}=\frac{3}{2}$ .

Substitute the calculated values into the general equation of a line to get the correct answer:

$y=\frac{3}{2}x-6$ .

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Question

In the standard (x, y) coordinate plane, a circle has the equation . At what points does the circle intersect the x-axis?

Answer

The generic equation of a circle is (x - x0)2 + (y - y0)2 = r2, where (x0, y0) are the coordinates of the center and r is the radius.

In this case, the circle is centered at the origin with a radius of 8. Therefore the circle hits all points that are a distance of 8 from the origin, which results in coordinates of (8,0) and (-8,0) on the x-axis.

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Question

What is the y-intercept of a line that passes through the point with slope of ?

Answer

Point-slope form follows the format y - y1 = m(x - x1).

Using the given point and slope, we can use this formula to get the equation y - 8 = -2(x + 5).

From here, we can find the y-intercept by setting x equal to zero and solving.

y - 8 = -2(0 + 5)

y - 8 = -2(5) = -10

y = -2

Our y-intercept will be (0,-2).

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Question

Given the linear equation below, what are the x- and y-intercepts_,_ respectively?

Answer

To find the x-intercept we will need to plug in zero for the y-value.

The x-intercept will be .

To find the y-intercept we will need to plug in zero for the x-value.

The y-intercept will be .

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Question

What are the y and x intercepts of the given equation, respectively?

y = 2x – 2

Answer

The equation is already in slope-intercept form. The y-intercept is (0, –2). Plug in 0 for y and we get the x intercept of (1, 0)

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Question

What is the x-intercept of the following line?

y = –3_x_ + 12

Answer

The x-intercept occurs when the y-coordinate = 0.

y = –3_x_ + 12

0 = –3_x_ + 12

3_x_ = 12

x = 12/3 = 4

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Question

What is the $\dpi{100} \small x$ -coordinate of the point in the standard $\dpi{100} \small (x,y)$ coordinate plane at which the two lines $\dpi{100} \small y=4x+8$ and $\dpi{100} \small y=3x-7$ intersect?

Answer

$\dpi{100} \small 4x+8=3x-7$

$\dpi{100} \small x+8=-7$

$\dpi{100} \small x=-15$

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Question

At what point do the lines $\small y=\frac{1}{4}x+7$ and $\small y=4x + 7$ intersect?

Answer

Short way:

The lines intersect somewhere because they have different slopes. Because they have the same y-intercept, they must intersect at that point.

Long way using substitution:

$\small y=\frac{1}{4}x+7$

$\small y-7 = \frac{1}{4}x$

$\small x = 4y - 28$

Plug this into $\small y=-4x+7$

$\small y=-4(4y-28)+7$

$\small y=-16y+112+7$

$\small 17y=119$

$\small y=7$

Find $\small x$

$\small x=4(7)-28=0$

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Question

What is the -intercept of the line in the standard coordinate plane that goes through the points and ?

Answer

The answer is .

The slope of the line is determined by calculating the change in over the change in .

The point-slope form of the equation for the line is then

. The -intercept is determined by setting and solving for . This simplifies to which shows that is the -interecept.

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Question

What are the and -intercepts of the line defined by the equation:

Answer

To find the intercepts of a line, we must set the and values equal to zero and then solve.

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Question

Find the -intercept(s) for the following equation:

Answer

To find the intercepts, is set equal to . This yields:

And finally $x=\pm3$

It is important to realize that both and must be included because is also equal to . Finally, these are put into their point forms, and .

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Question

What is the largest x-intercept of ?

Answer

To find the x-intercepts of an equation, you can just set the y value of the equation equal to zero. Thus you get, for our data:

Now, you can divide everything by to simplify your equation:

Luckily, this is an easy equation to factor:

Based on this, you know that the two intercepts must be where and . Thus, the largest intercept is .

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Question

What is the sum of the x-intercepts of ?

Answer

To find the x-intercepts of an equation, you can set its y value equal to zero. Thus, you get for our equation:

Now, factor out all common factors:

From this, you can further factor:

Thus, the x-intercepts of our equation are ,, and . The sum of these values is .

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Question

What is the $y\textup{ intercept}$ of the following equation: ?

Answer

The y-intercept is the constant at the end of the equation. Thus for our equation the y-intercept is 7

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Question

What is the -intercept of the following linear equation:
?

Give your answer as an ordered pair.

Answer

The x-intercept is the value of the linear equation with y = 0 (this means the line will be on the x-axis when y is zero).

Thus we plug 0 in for y and solve for x.

$\0 = -2x + 5 \ ewline 2x = 5 \ ewline x = \frac{5}{2}$ .

Now put it in an ordered pair, remember y = 0:
$\left(\frac{5}{2}, 0\right)$

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Question

What is the and intercepts of the linear equation given by:
?

Answer

To find the and intercept of a linear equation, find the points where and are equal to zero.

To do this, plug in zero for either variable and then solve for the other.

$\0 = 7x +8 ewline y = 7*0 + 8$

this yields:

$(0,8), \left(-\frac{8}{7},0\right)$

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Question

What is the and intercepts of the following linear equation:

Answer

To find the and intercepts of an equation, set each variable to zero (one at a time) and solve for the other variable.
$\0 = -8x + 3 ewline -3 = -8x ewline \frac{3}{8} = x$

Next, set to zero:

$\y = -8(0) +3 ewline y = 3$
Now put these two sets of points into two ordered pairs:

$\left(\frac{3}{8},0\right)$

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