Home / Digital SAT Math Practice Questions – Medium : Systems of two linear equations in two variables

# Digital SAT Math Practice Questions – Medium : Systems of two linear equations in two variables

## 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) $$9=x-\frac{14}{3}$$

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 $$x$$ ?

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

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. $$(15,66)$$

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

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

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 $$7.5$$.

$\boxed{B) \, 7.5}$

[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

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:
$p = 3$

$\boxed{D}$

[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?

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)

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

D) -11

$\begin{cases} 4x – 8y = -1 \\ x + 6y = -10 \end{cases}$

$\begin{cases} 4x – 8y = -1 \\ x + 2y = -10 \end{cases}$

$(4x+x) + (-8y +6y)=(-1) + (-10)$

$(5x) + (-2y)=(-11)$

[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

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

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 xy = 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? 1. $$y=2x+5$$ 2. $$y=\frac{1}{2}x+5$$ 3. $$y=\frac{1}{2}x-4$$ 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\$

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