ACT study guide Math: slopes
In the standard (x, y) coordinate plane, what is the slope of the line joining the points (3, 7) and (4, −8)?
(To find the slope of the line between any two points (x1, y1) and (x2, y2), you can use the equation (y2−y1)/(x2−x1). Therefore, when you have the points (3, 7) and (4, −8) it follows that the slope of the line joining these points is (−8−7)/(4−3)= )−15/1, or -15
What is the slope of the line given by the equation 21x − 3y + 18 = 0?
(To find the slope, convert the equation 21x − 3y + 18 = 0 to slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. To do so, first subtract 18 and 21x from both sides to get −3y = −21x − 18. Then divide both sides by −3 to get y = 7x + 6. Therefore the slope is) 7
What is the slope of the line y = −6?
Any line of the form y = c will have a slope of zero, since the y-coordinates will not change.) 0
Which of the following lines has a slope of zero?
Any line with a slope of zero will be parallel to the x-axis
What is the slope of the line 3x − 8y = 21?
(Solving for y, the line can be written as y=−21/8 + 3/8x. This is the slope-intercept form for the line, and the slope is the coefficient of x, or) 3/8.
What is the slope of any line parallel to the y-axis in the (x, y) coordinate plane?
(Any line parallel to the y-axis is a vertical line. Vertical lines have slopes that are undefined. Remember that the definition of slope is rise/run; vertical lines have no run and thus dividing rise by run is dividing by 0, making the quotient) undefined
The line passes through the points (1, 0) and (0, 2), so the slope is 2−0/0−1 = −2
What is the slope of the line in the following figure?
Which of the following equations represents the same line as the equation −2x + 6y = 14?
(Each of the lines in the answer choices is in slope-intercept form. Solving for y in the given equation will give us the equivalent equation in slope-intercept form. In this case, that would yield) y=1/3x + 7/3
In the standard (x, y) coordinate plane, if the x-coordinate of each point on a line is 5 more than half the y-coordinate, what is the slope of the line?
(If the x-coordinate of each point on a line is 5 more than half the y-coordinate, then x= y2+5. To find the slope of the line, solve for y and put the equation in slope-intercept form (y = mx + b, where m is the slope). To do so, first subtract 5 from both sides, then multiply the entire equation by 2 to get y = 2x − 10. The slope is )2
A system of linear equations is shown below. 4y=3x+12 −4y=−3x−8 Which of the following describes the graph of this system of linear equations in the standard (x, y) coordinate plane?
(To see the properties of each linear equation more clearly, convert each to slope-intercept form (y = mx + b, where m is the slope and b is the y-intercept). The equation 4y = 3x + 12 becomes y=3x/4+3 after you divide by 4. The equation −4y = − 3x − 8 becomes y=3x/4+2 after you divide by −4. It is now clear that the equations are lines with slope 3/4 , making them parallel lines with positive slope. However, since they have different y-intercepts, they are two distinct lines.) Two parallel lines with positive slope
Which one of the following lines has a negative slope?
(To solve this problem, first put the equations in the standard form, y = mx + b, where m is the slope. Answer choices A and B are already in the standard form, and each has a positive slope, so eliminate these choices. y + 1/2x = 1) y = −12x+1
What is the slope of a line that passes through the origin and the point (−6, 2) ?
(To solve this problem, recall that the formula for finding the slope of a line between the two points (x1, y1) and (x2, y2) is )y2−y1x2−x1. (Also recall that the origin lies at point (0, 0). Therefore, you know 2 points on this line: (−6, 2) and (0, 0). You can use either set of points as (x1, x2) and (y1, y2) in the slope formula, as long as you use them consistently within the formula, as follows: 2−0/−6−0 = 2/−6=)−1/3)
What is the slope of a line that passes through the points (−1/2,2) and (−1/4,1/4) in the (x, y) coordinate plane?
(m=1/4−2/−1/4−(−1/2)=1/4−8/4/−1/4+2/4=−7/4 / 1/4=) −7
(Because there is a right angle at S, the point T will lie along the line through S that is perpendicular to the segment RS⎯⎯⎯⎯⎯. To solve this problem, find the equation for the line through S that is perpendicular to the segment RS⎯⎯⎯⎯⎯ and try each answer choice to find one that lies on the line. Since the line is perpendicular to segment RS⎯⎯⎯⎯⎯, it will have a slope that is the negative reciprocal of the slope of RS⎯⎯⎯⎯⎯. Since slope is rise/run, the slope of RS⎯⎯⎯⎯⎯ is (3−2)/(6−2)=14. The slope of a line perpendicular to that is −4. Because a point and the slope of the line are known, the point-slope form of the equation can be utilized. A line through point (h, −k) with slope m has equation y − k = m(x − h). Thus the line through S (6, −3) that is perpendicular to the segment RS⎯⎯⎯⎯⎯ has equation y − 3 = −4(x − 6). Distributing and adding like terms, the result is y = −4x + 27. Of the answer choices, only the point (5, 7) falls on the line.)5, 7
The points R (2, 2) and S (6, 3) in the standard (x, y) coordinate plane below are 2 vertices of △RST, which has a right angle at S. Which of the following could be the third vertex, T ?
A line in the (x, y) coordinate plane has a slope of −5/4 and passes through the point (−3, 5). What is the y-intercept of this line?
Using the point-slope formula, y−5=−5/4(x−(−3)). This equation simplifies to y=−5/4x + 5/4
