SAT MAth Practice questions – all topics
- Algebra Weightage: 35% Questions: 13-15
- Linear equations in one variable
- Linear equations in two variables
- Linear functions
- Systems of two linear equations in two variables
- Linear inequalities in one or two variables
SAT MAth and English – full syllabus practice tests
[Calc] Question Medium
\[
6=\frac{2}{3}(x-7)
\]
Which equation has the same solution as the given equation?
A) \(9=x-14\)
B) \(9=\frac{2}{3} x-7\)
C) \(9=x-7\)
D) \(9=x-\frac{14}{3}\)
▶️Answer/Explanation
Ans:C
Given equation:
\[
6 = \frac{2}{3}(x – 7)
\]
To find an equivalent equation, start by clearing the fraction. Multiply both sides by \(\frac{3}{2}\):
\[
6 \cdot \frac{3}{2} = x – 7
\]
\[
9 = x – 7
\]
So, the equation equivalent to the given equation is:
\[
\boxed{9 = x – 7}
\]
[Calc] Question medium
$
\begin{aligned}
& 3 x+4 y=18 \\
& 2 x-4 y=17
\end{aligned}
$
The solution to the given system of equations is \((x, y)\). What is the value of \(x\) ?
▶️Answer/Explanation
Ans:7
To solve the given system of equations, we can use the method of elimination.
Given the system:
\[
\begin{align*}
3x + 4y &= 18 \\
2x – 4y &= 17
\end{align*}
\]
We can eliminate \(y\) by adding the equations together:
\[ (3x + 4y) + (2x – 4y) = 18 + 17 \]
\[ 3x + 2x = 35 \]
\[ 5x = 35 \]
\[ x = \frac{35}{5} \]
\[ x = 7 \]
So, the value of \(x\) is \(7\).
[Calc] Question medium
$
\begin{aligned}
& y=4 x+1 \\
& y=4 x+3
\end{aligned}
$
How many solutions does the given system of equations have?
A. Zero
B. Exactly one
C. Exactly two
D. Infinitely many
▶️Answer/Explanation
Ans:A
The given system of equations is:
\[
\begin{align*}
y &= 4x + 1 \\
y &= 4x + 3 \\
\end{align*}
\]
These are two parallel lines with the same slope \(4\). Since they have the same slope but different intercepts, they will never intersect. Therefore, the system has zero solutions, which corresponds to option A: Zero.
[No calc] Question medium
$
\begin{gathered}
y=4 x+6 \\
-5 x-y=21
\end{gathered}
$
What is the solution \((x, y)\) to the given system of equations?
A. \((-3,-6)\)
B. \(\left(-\frac{5}{3},-\frac{2}{3}\right)\)
C. \((3,18)\)
D. \((15,66)\)
▶️Answer/Explanation
Ans:A
To solve the system of equations:
\[
\begin{gathered}
y=4x+6 \\
-5x-y=21
\end{gathered}
\]
We can use substitution or elimination method. Let’s use the substitution method:
From the first equation, we have \(y = 4x + 6\). Substitute this expression for \(y\) into the second equation:
\[-5x – (4x + 6) = 21\]
\[-5x – 4x – 6 = 21\]
\[-9x – 6 = 21\]
Now, add 6 to both sides:
\[-9x = 27\]
Divide both sides by -9:
\[x = -3\]
Now, substitute \(x = -3\) back into one of the equations to find \(y\):
\[y = 4(-3) + 6\]
\[y = -12 + 6\]
\[y = -6\]
So, the solution to the system of equations is \((-3, -6)\), which corresponds to option A.
[No calc] Question medium
$
|x-10|=0
$
What are all possible solutions to the given equation?
A. -10
B. 0
C. 10
D. -10 and 10
▶️Answer/Explanation
Ans:C
The equation \(|x – 10| = 0\) implies that the expression inside the absolute value must be equal to 0 for the equation to be true. So,
\[x – 10 = 0\]
\[x = 10\]
Thus, the only possible solution is \(x = 10\), which corresponds to option C.
[Calc] Question Medium
\[
\begin{aligned}
& y=3 x+6 \\
& y=-3 x+9
\end{aligned}
\]
The solution to the given system of equations is \((x, y)\). What is the value of \(y\) ?
A) 15
B) 7.5
C) 1.5
D) 0.5
▶️Answer/Explanation
B
Given the system of equations:
\[ \begin{aligned} y &= 3x + 6 \\ y &= -3x + 9 \end{aligned} \]
To find the solution \((x, y)\), we can set the two equations equal to each other:
\[ 3x + 6 = -3x + 9 \]
Solve for \(x\):
\[ 3x + 3x = 9 – 6 \]
\[ 6x = 3 \]
\[ x = \frac{3}{6} = \frac{1}{2} \]
Now, substitute \(x = \frac{1}{2}\) into either of the original equations to find \(y\):
\[ y = 3\left(\frac{1}{2}\right) + 6 = \frac{3}{2} + 6 = \frac{15}{2} = 7.5 \]
Therefore, the value of \(y\) is \(7.5\).
So the answer is:
\[ \boxed{B) \, 7.5} \]
[Calc] Question Medium
\[
\begin{aligned}
& y=3 x+5 \\
& y=p x+8
\end{aligned}
\]
In the given system of equations, \(p\) is a constant. The system has no solution. What is the value of \(p\) ?
A) -3
B) \(-\frac{1}{3}\)
C) \(\frac{1}{3}\)
D) 3
▶️Answer/Explanation
D
The given system of equations is:
\[ y = 3x + 5 \]
\[ y = px + 8 \]
For the system to have no solution, the lines represented by these equations must be parallel. This means their slopes must be equal but their y-intercepts must be different.
The slope of the first equation is 3.
The slope of the second equation is \( p \).
Since the slopes must be equal for the lines to be parallel:
\[ p = 3 \]
So the answer is:
\[ \boxed{D} \]
[Calc] Question Medium
6x – y = -4
9x – y = -3
The solution to the given system of equations is (x, y) . What is the value of y?
▶️Answer/Explanation
6
To find the value of \(y\) in the system of equations:
\[
\begin{aligned}
6x – y &= -4 \\
9x – y &= -3
\end{aligned}
\]
1. Subtract the first equation from the second equation to eliminate \(y\):
\[
(9x – y) – (6x – y) = -3 – (-4)
\]
\[
9x – y – 6x + y = -3 + 4
\]
\[
3x = 1
\]
2. Solve for \(x\):
\[
x = \frac{1}{3}
\]
3. Substitute \(x\) back into the first equation to solve for \(y\):
\[
6\left(\frac{1}{3}\right) – y = -4
\]
\[
2 – y = -4
\]
\[
-y = -6
\]
\[
y = 6
\]
Thus, the value of \(y\) is:
\[ \boxed{6} \]
[No calc] Question medium
𝑦 = 7 − 4𝑥
15𝑥 − 4𝑦 = 3
What is the solution (x, y) to the given system of equations?
A. (-31, 131)
B. (-1, 11)
C. (1, 3)
D. (1, 11)
▶️Answer/Explanation
Ans: C
To find the solution \((x, y)\) to the given system of equations, we can use the substitution or elimination method.
Given equations:
1. \(y = 7 – 4x\)
2. \(15x – 4y = 3\)
Let’s substitute the expression for \(y\) from equation 1 into equation 2:
\[15x – 4(7 – 4x) = 3\]
Simplify the equation:
\[15x – 28 + 16x = 3\]
\[31x – 28 = 3\]
\[31x = 3 + 28\]
\[31x = 31\]
\[x = 1\]
Now, substitute \(x = 1\) into equation 1 to find \(y\):
\[y = 7 – 4(1)\]
\[y = 7 – 4\]
\[y = 3\]
So, the solution \((x, y)\) is \((1, 3)\), which corresponds to option C.
[Calc] Question Medium
4x – 8y= -1
x +6y =-10
The solution to the given system of equations is (x, y) . What is the value of 5x – 2y ?
A) 10
B) 9
C) -9
D) -11
▶️Answer/Explanation
D) -11
\[
\begin{cases}
4x – 8y = -1 \\
x + 6y = -10
\end{cases}
\]
2. Adding both equations:
\[
\begin{cases}
4x – 8y = -1 \\
x + 2y = -10
\end{cases}
\]
\[(4x+x) + (-8y +6y)=(-1) + (-10)\]
\[(5x) + (-2y)=(-11) \]
[Calc] Question medium
𝑦 = 3𝑥 + 9
𝑦 = −3𝑥 + 3
The solution to the given system of equations is (𝑥, 𝑦). What is the value of y
▶️Answer/Explanation
Ans: 6
Since both equations are equal to \(y\), we can set them equal to each other:
\[3x + 9 = -3x + 3\]
Now, let’s solve for \(x\):
\[3x + 3x = 3 – 9\]
\[6x = -6\]
\[x = -1\]
Now that we have \(x = -1\), we can substitute it into either of the original equations to find \(y= 3𝑥 + 9\).
\[y = 3(-1) + 9\]
\[y = -3 + 9\]
\[y = 6\]
So, the value of \(y\) is \(6\).
[Calc] Question Medium
The table shows the prices of 3 items in a certain store on January \(15,1913\).
On January 15, 1913, Ayana purchased eggs and potatoes for a total of \(\$ 8.80\). She purchased 24 eggs. Based on the prices in the table, how many pounds of potatoes did she purchase?
A) 3
B) 6
C) 22
D) 40
▶️Answer/Explanation
Ans: A
First, calculate the cost of the 24 eggs:
\[
\text{Cost of 24 eggs} = 2 \times \text{Cost of 12 eggs} = 2 \times \$0.37 = \$0.74
\]
Next, determine the remaining amount she spent on potatoes:
\[
\text{Remaining amount for potatoes} = \$0.80 – \$0.74 = \$0.06
\]
Now, find out how many pounds of potatoes she could buy with \$0.06:
\[
\text{Price of 1 pound of potatoes} = \$0.02
\]
So, the number of pounds of potatoes she purchased is:
\[
\text{Number of pounds of potatoes} = \frac{\$0.06}{\$0.02 \text{ per pound}} = 3 \text{ pounds}
\]
Thus, Ayana purchased 3 pounds of potatoes.
The correct answer is:
\[
\boxed{3}
\]
[Calc] Question medium
2x − 3y = 5
One of the two equations in a system is given. The system has an infinite number of solutions. Which equation could be the other equation in the system?
A) 4x − 6y = 10
B) 4x + 6y = 10
C) 2x − 3y = 10
D) 2x + 3y = 10
▶️Answer/Explanation
A) 4x − 6y = 10
Given the equation \(2x – 3y = 5\) and the fact that the system has an infinite number of solutions, we need to find the equation that corresponds to a line parallel to \(2x – 3y = 5\).
The equation of a line parallel to \(2x – 3y = 5\) will have the same slope but a different y-intercept.
Let’s rewrite the equation \(2x – 3y = 5\) in slope-intercept form:
\[
y = \frac{2}{3}x – \frac{5}{3}
\]
We can observe that the slope of this line is \(\frac{2}{3}\).
Among the given options, we need to find an equation with the same slope. The equation with the same slope is option B) \(4x + 6y = 10\).
Let’s rewrite \(4x + 6y = 10\) in slope-intercept form:
\[
6y = -4x + 10
\]
\[
y = -\frac{4}{6}x + \frac{10}{6}
\]
\[
y = -\frac{2}{3}x + \frac{5}{3}
\]
This equation has the same slope as the original equation, so it corresponds to a line parallel to \(2x – 3y = 5\), leading to an infinite number of solutions.
[Calc] Question medium
0.10x + 0.20y = 0.18(x + y)
The given equation represents a volume x, in gallons, of a 10% saltwater solution that will be mixed with a volume y, in gallons, of a 20% saltwater solution to produce an 18% saltwater solution. What volume, in gallons, of the 20% saltwater solution will be needed if 50 gallons of the 10% saltwater solution is used?
▶️Answer/Explanation
200
To solve this problem, we can use the given equation to find the volume of the \(20\%\) saltwater solution needed when \(50\) gallons of the \(10\%\) saltwater solution is used.
The given equation is:
\[ 0.10x + 0.20y = 0.18(x+y) \]
We are given that \(x = 50\) gallons (volume of the \(10\%\) saltwater solution).
We need to find \(y\), the volume of the \(20\%\) saltwater solution.
Substitute \(x = 50\) into the equation:
\[ 0.10(50) + 0.20y = 0.18(50 + y) \]
Simplify the equation:
\[ 5 + 0.20y = 9 + 0.18y \]
Subtract \(0.18y\) from both sides:
\[ 0.02y = 4 \]
Divide both sides by \(0.02\):
\[ y = \frac{4}{0.02} = 200 \]
So, 200 gallons of the \(20\%\) saltwater solution will be needed when 50 gallons of the \(10\%\) saltwater solution is used.
[Calc] Question Medium
The line with equation y =ax+ b, where a and bare constants, has a slope of -2 and passes through the point (3, 8) in the xy-plane. What is the value of b ?
▶️Answer/Explanation
14
Given that the line has a slope of -2 and passes through the point (3,8), we can use the point-slope form of a linear equation:
\[ y – y_1 = m(x – x_1) \]
where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.
\[ y – 8 = -2(x – 3) \]
Expanding and simplifying:
\[ y – 8 = -2x + 6 \]
\[ y = -2x + 14 \]
Comparing this with the general form \( y = ax + b \), we find that \( b = 14 \).
So, the value of \( b \) is \( \boxed{14} \).
[No calc] Question medium
x + 2y = 11
3x + 3y = 24
The solution to the given system of equations is the ordered pair (x , y). What is the value of x ?
▶️Answer/Explanation
5
We’re given the system of equations:
\[
\begin{align*}
x + 2y &= 11 \\
3x + 3y &= 24
\end{align*}
\]
To solve this system, we can use the method of substitution or elimination.
Let’s use the method of elimination. Multiply the first equation by 3 to make the coefficients of \(x\) in both equations equal:
\[
\begin{align*}
3x + 6y &= 33 \\
3x + 3y &= 24
\end{align*}
\]
Now, subtract the second equation from the first:
\[
\begin{align*}
(3x + 6y) – (3x + 3y) &= 33 – 24 \\
3x + 6y – 3x – 3y &= 9 \\
3y &= 9 \\
y &= 3
\end{align*}
\]
Now, substitute \(y = 3\) into either of the original equations. Let’s use the first one:
\[x + 2(3) = 11\]
\[x + 6 = 11\]
\[x = 11 – 6\]
\[x = 5\]
So, the solution to the system of equations is \(x = 5\).
[Calc] Question medium
x + y = 10
x − y = 4
The solution to the given system of equations is (x, y). What is the value of 2x?
▶️Answer/Explanation
14
Let’s solve the system of equations:
\[
\begin{cases}
x + y = 10 \\
x – y = 4
\end{cases}
\]
Adding the two equations, we get:
\[ (x + y) + (x – y) = 10 + 4 \]
\[ 2x = 14 \]
\[ x = 7 \]
So, if \(x = 7\), then \(2x = 2 \times 7 = 14\).
Thus, the value of \(2x\) is 14.
[Calc] Questions Medium
x+2y =10
2x- y = 5
The solution to the given system of equations is (x, y). What is the value of 3x + y ?
A) 5
B) 7
C) 13
D) 15
▶️Answer/Explanation
Ans: D
\[
\begin{aligned}
& x+2 y=10 \\
& 2 x-y=5
\end{aligned}
\]
We can use the method of substitution or elimination. Let’s solve it using elimination:
Multiply the first equation by 2:
\[2(x + 2y) = 2(10) \Rightarrow 2x + 4y = 20\]
Subtract the second equation from this new equation:
\[(2x + 4y) – (2x – y) = 20 – 5\]
\[2x + 4y – 2x + y = 15\]
\[5y = 15\]
\[y = 3\]
Substitute the value of \(y\) into either of the original equations to find \(x\). Let’s use the first equation:
\[x + 2(3) = 10\]
\[x + 6 = 10\]
\[x = 4\]
So, the solution to the system of equations is \(x = 4\) and \(y = 3\).
Now, we can find the value of \(3x + y\):
\[3x + y = 3(4) + 3 = 12 + 3 = 15\]
Therefore, the value of \(3x + y\) is \(\mathbf{15}\)
[Calc] Question Medium
$$
4 x-y=3
$$
One of the two equations in a linear system is given. The system has exactly one solution. Which equation could be the other equation in the system?
A) $-4 x+y=6$
B) $4 x-y=3$
C) $4 x+y=5$
D) $4 x-y=5$
▶️Answer/Explanation
C
[Calc] Question Medium
$$
\begin{aligned}
& 4 x+y=4 \\
& 8 x+y=5
\end{aligned}
$$
If $(x, y)$ is the solution of the system of equations above, what is the value of $x$ ?
▶️Answer/Explanation
$.25,1 / 4$
Question
\(y=\frac{1}{2}x+5\)
One of the two equations in a linear system is given. The system has no solution. Which equation could be the second equation in this system?
- \(y=2x+5\)
- \(y=\frac{1}{2}x+5\)
- \(y=\frac{1}{2}x-4\)
- \(y=-2x+5\)
▶️Answer/Explanation
C
Questions
If $4 x^2+b x+9=0$, where $b$ is a constant, has exactly one solution, what is a possible value of $b$ ?
A. 72
B. 36
C. 12
D. 6
▶️Answer/Explanation
Ans: C
Questions
$7 x-5 y=4$
$
4 x-8 y=9
$
If $(x, y)$ is the solution to the system of equations above, what is the value of $3 x+3 y$ ?
A. -13
B. -5
C. 5
D. 13
▶️Answer/Explanation
Ans: B
Questions
$$
\begin{aligned}
& .5 x+y=a \\
& -3 x-2 y=5
\end{aligned}
$$
In the system of equations above, $a$ is a constant. What is the $y$-value of the solution to the system in terms of $a$ ?
A. $\frac{-3 a-25}{7}$
B. $\frac{a-1}{7}$
C. $\frac{2 a+5}{7}$
D. $\frac{10 a+5}{7}$
▶️Answer/Explanation
Ans: A
Questions
$y=2 x+4$
$
y=(x-3)(x+2)
$
The system of equations above is graphed in the $x y$-plane. At which of the following points do the graphs of the equations intersect?
A. $(-3,-2)$
B. $(-3,2)$
C. $(5,-2)$
D. $(5,14)$
▶️Answer/Explanation
Ans: D
Questions
$(x+2)^2+(y-3)^2=40$
$
y=-2 x+4
$
Which of the following could be the $x$-coordinate of a solution to the system of equations above?
A. $\sqrt{7}$
B. $\frac{\sqrt{35}}{2}$
C. $\frac{6+2 \sqrt{34}}{5}$
D. $\frac{4+\sqrt{191}}{5}$
▶️Answer/Explanation
Ans: A
Questions
$8 x-2 x(c+1)=x$
In the equation above, $c$ is a constant. If the equation has infinitely many solutions, what is the value of $c$ ?
A. $\frac{3}{2}$
B. $\frac{5}{2}$
C. $\frac{7}{2}$
D. $\frac{9}{2}$
▶️Answer/Explanation
Ans: B
Question
$2 x+3 y=4 y=2 x$ If the ordered pair $(x, y)$ satisfies the system of equations above, what is the value of $x$ ?
▶️Answer/Explanation
Ans: $1 / 2, .5$