ACT study guide Math: Equation of a Line

Given that y−5=12x+1 is the equation of a line, at what point does the line cross the x-axis?

(A graph crosses the x-axis at the point when y = 0. Given that y−5=x2+1, let y = 0 such that −5=x2+1. Subtracting 1 from both sides yields −6=x2. Multiplying by 2 yields −12 = x.) -12

Which of the following equations represents a line with a negative y-intercept in the (x, y) coordinate plane?

(Dividing the constant on the right side of each equation by the coefficient of y will give the value of the y-intercept. This is because it will transform the equation into standard line-slope form. In answer choice A, the y-intercept is −10/2=−5. This is the only choice with a negative y-intercept.)

What are the slope and y-intercept, respectively, for the following algebraic expression for the line 6x + 3y = 12?

(First divide both sides by 3 to get 2x + y = 12. Arrange the equation in slope intercept form: y = −2x + 4. )Slope = −2, y-intercept = 4

For some real number n, the graph of the line y = (n + 1)x + 6 in the standard (x, y) coordinate plane passes through (4, 8). What is the value of n?

(Given that the graph of the line y = (n + 1)x + 6 in the standard (x, y) coordinate plane passes through (4, 8), plug the values of the point (4, 8) into the equation and solve for n. Substituting (4, 8) into y = (n + 1)x + 6 yields 8 = (n + 1)(4) + 6 = 4n + 10. To solve 8 = 4n + 10, subtract 10 from both sides and divide by 4 to get) n=−12.

A line in the (x, y) coordinate plane has a positive y-intercept c and a negative slope. Which of the following statements MUST be true about the x-intercept of this line?

(With a negative slope, the line will fall from left to right. Starting with the positive y-intercept, this leaves the only possibility for an x-intercept as being along the positive x-axis.) x > 0

In the (x, y) coordinate plane, what is the y-intercept of the line 5x + 3y = 8?

(convert 5x + 3y = 8 to the slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. To do so, first subtract 5x from both sides to get 3y = −5x + 8. Dividing the entire equation by 3 yields y=−5x3+83.) 8/3

How many ordered pairs (x, y) of real numbers will satisfy the equation 5x − 7y = 13?

(it defines a line. There are an infinite number of points in a line, there are an infinite number of ordered pairs (x, y) of real numbers that satisfy the equation) Infinitely many

In the standard (x, y) coordinate plane, what is the equation of the line that passes through the origin and the point (3, 4)?

(the formula for slope is (y2−y1)(x2−x1), where (x1,y1) and (x2,y2) are two given points on a line.point (3, 4) will have slope (4−0)/(3−0)=4/3.) y=4/3x

The x-intercept of a line in the (x, y) coordinate plane is −6. If the slope of this line is 5, which of the following is the equation of this line in slope-intercept form?

(the x-intercept is −6, so the line passes through the point (−6, 0) line:y − 0 = 5(x + 6) Simplification:) y = 5x + 30

If a line in the (x, y) coordinate plane has x-intercept (x, 0) and y-intercept (0, y) such that x > y, which of the following could be a graph of the line?

5, 2

What is the slope-intercept form of the line 5x + 8y = 11?

(Solving for y, 8y = 11− 5x, and )y = 11/8 − 5/8x

A line segment AB⎯⎯⎯⎯⎯ in the (x, y) coordinate plane has a midpoint that lies on the y-axis and is perpendicular to that axis. If the coordinates of point A are (−8, 6), which of the following could be the coordinates of point B?

(If the midpoint of line AB⎯⎯⎯⎯⎯ is on the y-axis, then the distance from A to the y-axis must be the same as the distance from B to the y-axis. Additionally, since the line must be perpendicular to the y-axis, the y-coordinate of the two points will be the same. The point A is 8 units from the y-axis, so the point B must be as well.) 8, 6

The lines will intersect where y = 8. Substitute this value of y into the given equation: 8=3/2x+1 which can be simplified to 7=3/2x. To solve for x, multiply both sides by 2/3: x = 2/3(7) = 14/3.

(In the following figure, the equation of line m is y= 3/2x + 1, and line n, which is parallel to the x-axis, intersects the y-axis at the point A (0, 8). If lines m and n intersect at point B, what is the x-coordinate of point B?)

In the standard (x, y) coordinate plane, at what point does the graph of the line y − 7x = −10 cross the y-axis?

(The point where the line crosses the y-axis is the y-intercept and can be found by putting the line in slope-intercept form (solving for y). This is written as y = mx + b, with m as the slope and b as the y-intercept. The resulting equation is y = −10 + 7x, and b = −10.) y = −10

In the (x, y) coordinate plane, what is the value of the x-intercept of a line that passes through the points (0, 5) and (1, 6)?

(The slope of the line would be m = 6−5/1−0 = 1, and because the line passes through (0, 5), the y-intercept is 5. Therefore, the equation of the line is y = x + 5. To find the x-intercept, let y = 0 and solve for x) -5

In the standard (x,y) coordinate plane, the y-intercept of the line 6x + 2y = 14 is?

(The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept. Put the equation given in the problem in the slope-intercept form: 6x + 2y = 14 2y = − 6x + 14 y = − 3x + 7 The y-intercept is )7

What is the slope-intercept form of 6x − 2y − 4 = 0?

(The slope-intercept form of the equation of a line states that y = mx + b. To find the slope-intercept form of the equation 6x − 2y − 4 = 0, you must isolate y on the left side of the equation, as follows: 6x−2y−4=0 −2y−4=−6x −2y=−6x+4 y=3x−2 If you selected answer choice D, you probably forgot to switch the signs when dividing by −2. It is crucial to multiply all terms on both sides of the equation to arrive at a correct answer.) y=3x−2

Which of the following equations represents the same line in the (x, y) coordinate plane as y4+x=−12?

(This is point slope form question. 1.Subtract both sides y4=−12−x 2.multiply the sides by 4) y = −2 − 4x

What is the slope-intercept form of 9x + 3y − 6 = 0?

(To find the slope-intercept (y = mx + b) form of the equation 9x + 3y − 6 = 0, you could first add 6 and subtract 9x from both sides of the equation to get 3y = −9x + 6. Then, multiply both sides by 1/3 to get) y = −3x + 2

What is the x-coordinate of the point in the standard (x, y) coordinate plane at which the two lines y = −2x + 7 and y = 3x − 3 intersect?

(To find the x-coordinate where the lines with equations y = −2x + 7 and y = 3x − 3 intersect, set −2x + 7 equal to 3x − 3 and solve for x: −2x + 7 = 3x − 3 −5x + 7 = −3 −5x = −10) x = 2

In the (x, y) coordinate plane, what is the x-intercept of the line y=−1/4x + 2?

(To find the x-intercept, let y = 0, and solve for x. The equation 0 = −1/4x+2 is equivalent to the equation −2 = −1/4x, which has a solution of x = )8

In the standard (x, y) coordinate plane, what is the y-intercept of the line given by the equation 3x+5y=8?

(To solve, convert the equation of the line to slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept). If 3x + 5y = 8, then: 3x+5y−3x = 8−3x (5y)/5 = (−3x+8)/5 y = −3x/5 + 8/5 Since the equation y = −3x/5 + 8/5 is in slope-intercept form, the y-intercept is )8/5