Heart of algebra

Heart of algebra is one of the 4 sections on the SAT Math Test. It has 19 questions. After careful research, I put these 31 questions together to help you prepare for the test.

Question 1 : Easy heart of algebra question

Solve 2x - 6 > 15 for x

Question 2

If  x  = z2 and z = 4/k, what is the value of x when k = 1/2

Question 3

In the equation below, a is a constant. For what value of a does the equation have infinitely many solutions?

2 + 8x - 6 = (2a + 3)x - 4

Question 4

P  = 2.6t + 312

The amount of money a car dealer made between 1995 and 2006 is modeled by the equation above, where P is the money the company made, in millions of dollars, and t is the year. Assuming 1995 is t = 0, what does 2.6 mean in the equation?

A. The amount of money was 2.6 million in 1995
B. The amount of money was 2.6 million in 2006
C. The amount of money the company made increased a total of 2.6 million between 1995 and 2006.
D. The amount of money the company made increased each year by 2.6 million between 1995 and 2006.

Question 5

A landscaper plans his time carefully to decide the number of hours, x, he will spend on mowing each week for the year of 2019. He expects to spend an additional 600 hours in 2019 planting flowers, digging, and trimming. If the landscaper is available to work 2160 hours in 2019, find an equation that describes his situation.

Question 6

y = 5x
x = 5y

How many solutions does the system above have ?

A.        0          B.       1.         C.         2.        D.        Infinitely many

Question 7 : Tricky heart of algebra question.

A college bookstore charges 80 dollars for a yearly membership. The first two books are free with membership, and any book after that cost 5.60 to include tax. How much money x does a student spend after buying y books and a yearly membership?

A. y = 80 + (x - 2) 5.60
B. x = (y - 2) 5.60
C. x = 80 + (y - 2) 5.60
D. y = (x - 2) 5.60

Question 8 :  Easy heart of algebra question

If x / 15 = (y - 2) / 3, what is x in terms of y?

Question 9

John bought a car at a dealer that gave a 25% discount off the original price. After paying s sales tax of 6% on the discount price, what is the original price in terms of x if the total amount he paid to the cashier is x?

Question 10

A video game company produced 4 millions of a specific game in 2015. Because the game became very popular, the company decided to produce 3 millions of that game every 2 years. At this rate, how many games will be produced in 2033?

Question 11

To promote his tutoring business, a math tutor charges only 40 dollars for the first 5 hours. However, after the first 5 hours, he charges 60 dollars per hour. Which of the following expresses the amount, A, in dollars, the math tutor charges if students must purchase at least 6 hours of tutoring?

A) A = 40X + 60
B) A = 40X + 260
C) A = 60X + 40
D) A = 60X - 260

Question 12

4 ( 1/3 - x ) = 5/4 + 6x

What is the solution of the equation above?

Question 13

-3 (5x - 100)  = 12 (-100 + 5x)

What is the solution of the equation above?

Question 14

In the system of equations below, a and b are constants. If the system has infinitely many solutions, what is the value of a?

5s - 6t = a

-10s + bt = -8

Question 15

x - 4y = 10

5x - 20y = 3

How many solutions (x, y) are there to the system of equation above?
A)        zero        B)         One        C)       Two        D)        More than two

Question 16

The graph of E is shown in the x-y plane below. Find an equation of a line that is perpendicular to line E and passing through the points (-1,0)


Question 17

Each week, Jetser rides his bike at 5 miles per hour and walks at 3 miles per hour. Jetser's goal is to ride his bike and walk a total of at least 7 miles in no more than 2 hours. If Jetser walks w miles and bikes b miles, which of the following systems of linear inequalities represents Jetser's goal?


w + b ≥ 7
b / 5 + w / 3 ≤ 2

w + b ≤ 7
b / 5 + w / 3 ≥ 2

w + b ≥ 7
5b + 3w ≤ 2

w + b ≤ 2
5b + 3w ≥ 7

More heart of algebra questions

Question 18

-2y = 8x + 2

3( 4 x + 1) = 2y - 10

What is the solution (x , y) to the system of equations above?

Question 19

On a Monday morning, an auto repair shop has done p oil change each hour for 6 hours while another has done n each hour for 4 hours. What is the total number of oil change done by both auto repair shops on that Monday?

Question 20 : Easy heart of algebra question

Solve the following absolute value equation for x

|6x - 4| = 14

Question 21

Beach chair rental cost

The graph above displays the total cost, C, in dollars, of renting a beach chair for h hours. Which of the following show the relationship between h and C?

A) C = 40h + 3

B) C = (40/3)h + 20

C) C = 3h + 40

D) h = 40c

Question 22

Before attending a basketball game, adults and children paid a total of 500 dollars. Two adults brought 6 children and the remaining adults brought 3 children each. If tickets were 25 dollars for each child and 40 dollars for each adults, how many adults and children attended the play?

A. 6 adults and 11 children
B. 5 adults and 12 children
C. 4 adults and 14 children
D. 3 adults and 10 children

Question 23

2x - 3y = 15
4x - ay = 6

Consider the system of linear equations above. Which value of a will result in a system of equations with no solutions?

Question 24 : Tricky heart of algebra question

A man wants to spend less than 40 minutes packing his s shoes and his p pants. The time he is going to spend packing his pants is at least as many as the time he is going to spend packing his shoes. Find a system of inequalities to model the situation above?

Question 25

Which of the following equations represent the line graphed below?

A. 6x - 2y = 6
B. 6x - 6y = 2
C. 6y - 6x = 2 
D. 6y - 2x = 6

Graph of a linear equation

Question 26

The amount of gas, g, in gallons in a car after the car has driven m miles can be modeled with the following equation. What is the meaning of the 25 in the equation?

g = 18 - m/25

A. The car can use 25 gallons of gas per mile
B. The car can travel 25 miles to the gallon
C. The car can hold 25 gallons of gas.
D. The car can drive 25 miles on a tank of gas.

Question 27

x - 6 = 10
y / 3 =1 + x / 2

Using the system of equations above, what is the value of x2 - y2 ?

Question 28

Each two dozen lemonades costs a store 13 dollars to make. The store owner plan to sell each lemonade for 2.25 dollars. Which expression shows the store owner's profit after selling x dozens lemonades?

A) 27x - 6.5x
B) 27x + 6.5x
C) x / 27 - 6.5x
D) 2.25x + 6.5x

Question 29

This month, Julian can spend at the most 40 dollars on toys using his monthly allowance. He already spend 15 dollars on a teddy bear. He now wants to buy sets of legos to play with his friends. If each set of legos costs 3 dollars, which of the following inequalities best models the situation describes above?

A) 3x + 15 ≤ 45
B) 3x - 15 ≤ 45
A) 3x + 15 ≥ 45
B) 3x - 15 ≥ 45

Question 30 : Tricky heart of algebra question.

A man needs to assemble 400 computers before a company can sell them. It takes the man 8 days to put the parts together for 20 of the computers and a minimum of 2 days to install all the software for these 20 computers. If n represents the number of computers the man can assemble in 60 days, find an inequality that best models the situation describe above?

Question 31

In order to help a young man save money for his trip to Italy, his parents gave him 100 on May 1, 2015. Then, they promised to give him 50 dollars at the beginning of each month. Which graph represents the total amount of money the boy will save after m months?

Graph of a linear equation

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