Unit 1
Evaluate f(x)=7x -5 when x=2.
ANSWER: f(2)=9
WORK:
f(2)=7(2)-5
f(2)=14-5
f(2)=9
Simplify (x2 + 5x - 7) + (-6x2 + 2x + 3)
ANSWER: -5x2 + 7x -4
WORK:
(x2 + 5x - 7) + (-6x2 + 2x + 3)
x2 - 6x2 + 5x+2x -7+3
-5x2 + 7x -4
Identify the slope and y-intercept of the linear function y=3x + 5
y =mx+b
slope = m = 3/1 = rise 3 / run 1
y-intercept = b = 5 or the point (0,5)
What is the domain of the following function: y+2= -3(x-1)
The domain is all real numbers.
Solve the following equation for x:
3(x+5) - 2x = 18
ANSWER: x = 3
WORK:
3x + 15 - 2x = 18
3x - 2x + 15 = 18
1x + 15 = 18
-15
x = 3
Evaluate the following expression:
(100)2/3
ANSWER: 10 cube root(10)
WORK:
cube root(100)2
cube root(100 * 100)
cube root(10 * 10 * 10 * 10)
10 cube root(10)
Simplify (5x3y2)(-2xy4)
ANSWER: -10x4y6
WORK:
(5)(-2)x3xy2y4
-10x3+1y2+4
-10x4y6
What is the slope of a line passing through the points (1, 5) and (-2, 3) ?
ANSWER: m = 2/3
WORK:
slope = m = rise /run = y2 - y1 / x2 -x1
(1,5) = (x1,y1) and (-2,3) = (x2, y2)
m = 3-5 / -2-1
m = -2 / -3
m = 2/3
What is the x-intercept of y=-2x+8
ANSWER: The x-intercept is 4, or the point (4,0) .
WORK:
x-intercepts have y=0
0 = -2x + 8
-8
-8 = -2x
/ -2
-8/-2 = -2/-2 x
4 = 1x
4 = x
The x-intercept is 4, or the point (4,0) .
Solve for m: 4(3m + 1) = -5(m - 2) + 45
ANSWER: m = 3
WORK:
4(3m + 1) = -5(m - 2) + 45
12m + 4 = -5m + 10 + 45
12m + 4 = -5m + 55
+5m
17m + 4 = 55
-4
17m = 51
/17
m = 3
Evaluate h(-7), when h(t)= 3t2 -5t + 1
ANSWER: h(-7) = 183
WORK:
h(-7) = 3(-7)2 -5(-7) + 1
h(-7) = 3(49) -5(-7) + 1
h(-7) = 147 + 35 + 1
h(-7) = 183
Simplify (x-7)(x+3)
ANSWER: x2 - 4x -21
WORK:
(x-7)(x+3)
(x * x) + (x * 3) + (-7 * x) + (-7 * 3)
(x2) + (3x) + (-7x) + (-21)
x2 + (-4x) + (-21)
x2 - 4x -21
Convert the following into slope-intercept form: 5y+2x = 35
ANSWER: y = -2/5 x + 7
WORK:
5y + 2x = 35
-2x
5y = -2x + 35
/5
y = -2/5 x + 35/5
y = -2/5 x + 7
Write a linear function parallel to y = 3x - 1 that goes through the point (-8, 2).
ANSWER: y - 2 = 3(x + 8)
WORK:
Parallel lines have the same slope
m = 3 , (-8,2)=(x1,y1)
y - y1= m(x - x1)
y - 2 = 3(x - -8)
y - 2 = 3(x + 8)
Given the equation Ax + By = C, solve for y.
ANSWER: y = (-Ax + C) / B
WORK:
Ax + By = C
-Ax
By = -Ax + C
/B
y = (-Ax + C) / B
Rewrite the following in radical form:
95/2
ANSWER: 243
WORK:
square root(9)5
square root(9 * 9 * 9 * 9 *9)
9 * 9 square root(9)
81 square root(3 * 3)
81 * 3
243
Simplify (2x + 6y)(x - 8y)
ANSWER: 2x2 - 10xy - 48y2
WORK:
(2x + 6y)(x - 8y)
(2x * x) + (2x * -8y) + (6y * x) + (6y * -8y)
(2x2) + (-16xy) + (6xy) + (-48y2)
2x2 + (-10xy) + (-48y2)
2x2 - 10xy - 48y2
What is the equation of a line with slope of -1/4 that passes through the point (2, -5) ?
ANSWER: y + 5 = -1/4(x-2)
WORK:
given a slope and a point, use point-slope form y-y1 = m(x-x1)
slope = m = -1/4 and point (2,-5) = (x1,y1)
y - y1 = m(x-x1)
y - -5 = -1/4 (x-2)
y + 5 = -1/4(x-2)
Write an equation for a line that passes through the point (1,-4) and is perpendicular to the line y + 2 = -2/3 (x - 5)
ANSWER: y + 4 = 3/2 (x - 1)
WORK:
Perpendicular lines have opposite reciprocal slopes
given m = -2/3 , opp rec m= 3/2
(1, -4) = (x1,y1)
y - y1 = m(x - x1)
y - -4 = 3/2 (x - 1)
y + 4 = 3/2 (x - 1)
Solve the following system of equations:
x = 2y
-3x + y = -15
ANSWER: (6, 3)
WORK:
*If you solve by substitution*
-3(2y) + y = -15
-6y + y = -15
-5y = -15
/-5
y = 3
x = 2(3)
x = 6
(6, 3)
Rewrite the following as a single power of 5 :
(25)3/4 * (5)-1/3
ANSWER: (5)7/6
WORK:
(25)3/4 * (5)-1/3
((5)2)3/4 * (5)-1/3
(5)2*3/4 * (5)-1/3
(5)6/4 * (5)-1/3
(5)3/2 * (5)-1/3
(5)3/2 + -1/3
(5)3/2 - 1/3
(5)9/6 - 2/6
(5)7/6
Divide the following: (15x5 +6x4 - 21x2) / (3x2)
ANSWER: 5x3 + 2x2 -7
WORK:
(15x5 +6x4 - 21x2) / (3x2)
(15x5 / 3x2) + (6x4 / 3x2) - (21x2 / 3x2)
(5x5-2) + (2x4-2) - (7x2-2)
(5x3) + (2x2) -(7x0)
5x3 + 2x2 - (7*1)
5x3 + 2x2 -7
Convert the following into standard form:
y - 1 = -1/3 (x+8)
ANSWER: x + 3y = -5
WORK:
y - 1 = -1/3 (x+8)
*3
3 (y - 1) = 3(-1/3)(x+8)
3y - 3 = (-3/3)(x+8)
3y - 3 = -1(x+8)
3y - 3 = -1x - 8
+3
3y = -1x - 5
+1x
3y + 1x = -5
3y + x = -5
x + 3y = -5
Write the equations for line k that passes through (8,11) and is perpendicular to line q that passes through (3, -1) and (-2, 7) .
ANSWER: y - 11 = 5/8 (x - 8)
WORK:
Start by finding the slope of line q.
(3, -1) = (x1,y1) and (-2, 7)=(x2,y2)
m = y2-y1 / x2-x1
m = 7- -1 / -2 -3
m = 7 + 1 / -2 -3
m =8/-5 = -8/5
perpendicular = opp rec m = 5/8 for line k
(8,11) = (x1,y1) for line k
y - y1 = m(x - x1)
y - 11 = 5/8 (x - 8)
Solve the following system of equations:
2x + 2y = 16
3x - 4y = 10
ANSWER: (6,2)
WORK:
*If you solve by elimination*
2 ( 2x + 2y = 16) >>> 4x + 4y = 32
4x + 4y = 32
3x - 4y = 10
add together
7x + 0y = 42
7x = 42
/7
x = 6
2(6) + 2y = 16
12 + 2y = 16
-12
2y = 4
/2
y = 2
(6,2)