SAT Math - Algebra Domain Test
Calculator is allowed on all questions. Figures are not necessarily drawn to scale.
Question 1
If \(4x+7=31\), what is the value of \(x\)?
Show solution
Subtract 7 from both sides: \(4x=24\).
Divide by 4: \(x=24/4=6\).
Answer: B
Question 2
A gym charges a one-time registration fee of \(12\) plus \(3\) for each class attended. If a member pays \(30\) total, how many classes did the member attend?
Show solution
Let \(c\) be the number of classes: then \(12+3c=30\), so \(3c=18\) and \(c=6\).
Answer: C
Question 3
Which inequality is equivalent to \(5x-4<16\)?
Show solution
Add 4 to both sides to get \(5x<20\): dividing by 5 gives \(x<4\).
Answer: A
Question 4
The points \((2,5)\) and \((6,13)\) lie on a line. What is the slope of the line?
Enter your answer:
Show solution
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
Substitute the two points: \(m=\frac{13-5}{6-2}=\frac{8}{4}=2\).
Answer: 2
Question 5
The formula \(P=2L+2W\) gives the perimeter of a rectangle. Which equation expresses \(L\) in terms of \(P\) and \(W\)?
Show solution
From \(P=2L+2W\), subtract \(2W\): \(P-2W=2L\).
Divide by 2: \(L=\frac{P-2W}{2}\).
Answer: D
Question 6
An arithmetic sequence begins \(11,15,19,23,\ldots\) What is the 10th term?
Show solution
For an arithmetic sequence, use \(a_n=a_1+(n-1)d\).
Here \(a_1=11\) and the common difference is \(d=4\).
The 10th term is \(a_{10}=11+(10-1)(4)=11+36=47\).
Answer: B
Question 7
The graph of a line is shown. What is the slope of the line?
Show solution
Use the slope formula \(m=\frac{y_2-y_1}{x_2-x_1}\).
The line passes through \((0,3)\) and \((6,0)\), so \(m=\frac{0-3}{6-0}=-\frac{1}{2}\).
Answer: C
Question 8
The system \(x+y=11\) and \(x-y=3\) has solution \((x,y)\). What is the value of \(x\)?
Enter your answer:
Show solution
Add the equations: \(2x=14\), so \(x=7\).
Answer: 7
Question 9
At a school play, adult tickets cost \(8\) and student tickets cost \(5\). A total of 40 tickets were sold for \(260\). How many adult tickets were sold?
Show solution
Let \(a\) be adult tickets and \(s\) student tickets: then \(a+s=40\) and \(8a+5s=260\).
Substitute \(s=40-a\): \(8a+5(40-a)=260\). \(3a=60\) and \(a=20\).
Answer: B
Question 10
Which equation represents the line through \((4,-1)\) that is parallel to \(3x-2y=10\)?
Show solution
Rewrite the given line as \(y=\frac{3}{2}x-5\).
A parallel line has slope \(\frac{3}{2}\).
Using \((4,-1)\) gives \(-1=6+b\), so \(b=-7\).
Answer: A
Question 11
Which compound inequality is equivalent to \(2<3x-4\le 11\)?
Show solution
Add 4 throughout: \(6<3x\le15\).
Divide throughout by 3: \(2<x\le5\).
Answer: C
Question 12
A line has slope \(-3\) and passes through \((5,4)\). What is the \(y\)-intercept of the line?
Enter your answer:
Show solution
Use \(y=-3x+b\): substituting \((5,4)\) gives \(4=-15+b\), so \(b=19\).
Answer: 19
Question 13
For what value of \(k\) does the system \(2x+3y=7\) and \(6x+9y=k\) have infinitely many solutions?
Show solution
The second equation must be exactly three times the first.
Tripling \(2x+3y=7\) gives \(6x+9y=21\), so \(k=21\).
Answer: D
Question 14
A tank contains 480 liters of water and drains at a constant rate. After 6 minutes, 390 liters remain. Which equation gives the amount \(A\), in liters, remaining after \(t\) minutes?
Show solution
The amount decreases by \(480-390=90\) liters in 6 minutes, a rate of \(15\) liters per minute.
Starting from 480 gives \(A=480-15t\).
Answer: A
Question 15
The graph of a system of two linear equations is shown. How many solutions does the system have?
Show solution
The two lines are distinct and parallel.
They never intersect.
Therefore, the system has \(0\) solutions.
Answer: A
Question 16
The equation \(7x+4y=52\) is satisfied by the point \((4,y)\). What is the value of \(y\)?
Enter your answer:
Show solution
Substitute \(x=4\): \(7(4)+4y=52\).
Then \(28+4y=52\), so \(4y=24\) and \(y=6\).
Answer: 6
Question 17
The equation \(6x+9y=15\) is one equation in a system of two linear equations. The system has no solution. Which equation could be the second equation?
Show solution
Divide the first equation by 3 to get \(2x+3y=5\): a system has no solution when the second line has the same slope but a different intercept.
\(4x+6y=13\) has proportional \(x\)- and \(y\)-coefficients but a nonproportional constant.
it is parallel and distinct.
Answer: D
Question 18
A rental company charges a fixed fee plus an hourly rate. A 3-hour rental costs \(74\), and a 7-hour rental costs \(122\). What is the fixed fee, in dollars?
Enter your answer:
Show solution
The hourly rate is \(\frac{122-74}{7-3}=12\) dollars per hour.
If the fixed fee is \(f\), then \(74=f+3(12)\), so \(f=38\).
Answer: 38
Question 19
Line \(m\) passes through \((2,9)\) and \((8,21)\). Line \(n\) is perpendicular to line \(m\) and passes through \((6,2)\). What is the \(y\)-intercept of line \(n\)?
Show solution
The slope of line \(m\) is \(\frac{21-9}{8-2}=2\).
The slope of a perpendicular line is \(-\frac{1}{2}\).
For line \(n\), \(y=-\frac{1}{2}x+b\).
Using \((6,2)\) gives \(2=-3+b\), so \(b=5\).
Answer: B
Question 20
The system \(3(x-2)+2(y+1)=19\) and \(5(x-2)-2(y+1)=13\) has a solution \((x,y)\). What is the value of \(6(x-2)\)?
Show solution
Let \(u=x-2\) and \(v=y+1\): the system becomes \(3u+2v=19\) and \(5u-2v=13\).
Adding gives \(8u=32\), so \(u=4\).
Therefore \(6(x-2)=6u=24\).
Answer: C