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

# Digital SAT Math Practice Questions – Advanced : 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 Hard

\begin{aligned} & -8 x-24=10 y \\ & 15 y=6-18 x \end{aligned}

The solution to the given system of equations is $$(x, y)$$. What is the value of $$x$$ ?

Ans: 7

To solve the system of equations:
$\begin{cases} -8x – 24 = 10y \\ 15y = 6 – 18x \end{cases}$

We can first isolate $$y$$ in terms of $$x$$ from the second equation and divide by 3 both numerator and denominator:
$15y = 6 – 18x \\ y = \frac{6 – 18x}{15} = \frac{2 – 6x}{5}$

Now, we can substitute this expression for $$y$$ into the first equation:
$-8x – 24 = 10\left(\frac{2 – 6x}{5}\right)$
$-8x – 24 = 4 – 12x$

Solve for $$x$$:
$-8x + 12x = 2 + 24$
$4x = 28$
$x = \frac{28}{4}$
$x = 7$

[No calc]  Question   Hard

2x-2y=2

One of the two linear equations in a system is given. The system has exactly one solution. Which equation could be the second equation in this system?

A. -8x+8y =- 3

B. 3x-3y=8

C. -10x+8y=5

D. -x+y =- 1

Ans: C

To find the second equation that forms a system with the given equation $2x – 2y = 2……(1)$, and has exactly one solution, we need to find an equation that is not a multiple of the given equation and intersects it at a single point.

A. -8x + 8y = -3
If we multiply the given equation(1) by -4, we get $-8x+8y = -8$.
This equation is parallel to $-8x + 8y = -3$, which means they have the same slope but different y-intercepts.
Parallel lines do not intersect, so this option is incorrect.

B. 3x – 3y = 8
If we multiply the given equation(1) by 3/2, we get 3x – 3y = 3.
This equation is parallel to 3x – 3y = 8, which means they have the same slope but different y-intercepts.
Parallel lines do not intersect, so this option is incorrect.

C. -10x + 8y = 5
If we multiply the given equation(1) by -5, we get -10x + 10y = -10.
This equation is not parallel to -10x + 8y = 5, which means they have different slopes and intersect at a single point.
This option is correct.

D. -x + y = -1
If we multiply the given equation(1) by -1/2, we get -x + y = -1.
This equation is coincident to -x + y = -1, which means they have the same slope .
Coincident lines do not intersect, so this option is incorrect.

Therefore, the second equation in the system C. -10x + 8y = 5 .

[Calc]  Question  Hard

One of the two linear equations in a system is −6𝑥 + 7𝑦 = −6. The system has no solution. Which equation could be the second equation in this system?

A. 6𝑥 − 7𝑦 = 0
B. $$-\frac{21x}{4}+\frac{49y}{8}=-\frac{21}{4}$$
C. $$-\frac{21x}{4}-14y=0$$
D. 6𝑥 − 7𝑦 = 6

Ans: A

When a system of linear equations has no solution, it means the lines represented by the equations are parallel, and thus their slopes are equal but their $$y$$-intercepts are not.

The given equation is $$-6 x+7 y=-6$$. To find the slope of this line, let’s rewrite it in slope-
\begin{aligned} & \text { intercept form }(y=m x+b) \text { : } \\ & -6 x+7 y=-6 \\ & 7 y=6 x-6 \\ & y=\frac{6}{7} x-\frac{6}{7} \end{aligned}

So, the slope of this line is $$m=\frac{6}{7}$$.

Option A and D having Equal slope but in D it is coincident line ( infinite no. of solution) as both have -6 intercept.

So correct option is A.

[No calc]  Question  Hard

−3x + 4y = 4

4x − 3y = 0.5

The solution to the given system of equations is the ordered pair (x,y) What is the value of y ?

2.5

To find the value of $$y$$ in the system of equations:
\begin{aligned} -3x + 4y &= 4 \quad \text{(1)} \\ 4x – 3y &= 0.5 \quad \text{(2)} \end{aligned}

Solve one of the equations for one variable. From equation (2):
$4x – 3y = 0.5$
$4x = 3y + 0.5$
$x = \frac{3y + 0.5}{4}$

Substitute $$x = \frac{3y + 0.5}{4}$$ into equation (1):
$-3\left(\frac{3y + 0.5}{4}\right) + 4y = 4$
$-\frac{9y + 1.5}{4} + 4y = 4$
$-\frac{9y + 1.5}{4} + \frac{16y}{4} = 4$
$\frac{7y – 1.5}{4} = 4$

Multiply both sides by 4 to clear the fraction:
$7y – 1.5 = 16$

$7y = 17.5$

Divide by 7:
$y = 2.5$

[Calc]  Question   Hard

\begin{aligned} & y=\frac{3}{2} x-\frac{1}{2} \\ & y=\frac{k}{3} x+\frac{1}{3} \end{aligned}

In the system of equations above, $$\mathrm{k}$$ is a constant. If the system has no solutions, what is the value of $$\mathrm{k}$$ ?

Ans:4.5 Or 9/2

Given the equations:
$y = \frac{3}{2}x – \frac{1}{2}$
$y = \frac{k}{3}x + \frac{1}{3}$

The slopes of the lines are $$\frac{3}{2}$$ and $$\frac{k}{3}$$. For the lines to be parallel and have no solutions, these slopes must be equal:
$\frac{3}{2} = \frac{k}{3}$

Cross multiply and solve for $$k$$:
$3 \times 3 = 2 \times k$
$9 = 2k$
$k = \frac{9}{2}$

So, the value of $$k$$ that results in no solutions is $$k = \frac{9}{2}$$.

[Calc]  Question  Hard

5x+4y=44

3x-7y=17
The solution to the given system of equations is (x,y). What is the value of  8x-3y ?

Ans: 61

we can use either the substitution method or the elimination method. Here, we’ll use the elimination method.

First, let’s eliminate one of the variables. Multiply the first equation by 3 and the second equation by 5:

\begin{aligned} 15x + 12y &= 132, \\ 15x – 35y &= 85. \end{aligned}

Now subtract the second equation from the first:

$(15x + 12y) – (15x – 35y) = 132 – 85, \\ 47y = 47, \\ y = 1.$

Now substitute $$y = 1$$ back into the first equation to find $$x$$:

$5x + 4(1) = 44, \\ 5x + 4 = 44, \\ 5x = 40, \\ x = 8.$

So, the solution to the system is $$(x, y) = (8, 1)$$.

Now we need to find the value of $$8x – 3y$$:

$8x – 3y = 8(8) – 3(1) = 64 – 3 = 61.$

[Calc]  Question  Hard

When Karina walks from home to work, she burns 5.3 calories per minute, and when she rides her bike from home to work she burns 6.4 calories per minute. If Karina spends a total of 6 hours walking and bicycling from home to work in a week and burns a total of 1941 calories doing these activities, how many minutes does she spend bicycling?

30

Let $$w$$ be the number of minutes Karina spends walking and $$b$$ be the number of minutes she spends bicycling.

According to the given information:
1. Karina burns $$5.3$$ calories per minute while walking.
2. Karina burns $$6.4$$ calories per minute while bicycling.
3. Karina spends a total of $$6$$ hours walking and bicycling in a week, which is $$6 \times 60 = 360$$ minutes.
4. Karina burns a total of $$1941$$ calories doing these activities.

We can set up a system of equations based on the given information:

$\begin{cases} 5.3w + 6.4b = 1941 \\ w + b = 360 \end{cases}$

We need to solve this system of equations to find the value of $$b$$, the number of minutes Karina spends bicycling.

Using substitution or elimination to solve the system.

Multiply the second equation by $$5.3$$ to eliminate $$w$$:

$5.3(w + b) = 5.3 \times 360$
$5.3w + 5.3b = 1908$

Now, subtract this equation from the first equation:

$(5.3w + 6.4b) – (5.3w + 5.3b) = 1941 – 1908$
$6.4b – 5.3b = 33$
$1.1b = 33$
$b = \frac{33}{1.1}$
$b = 30$

So, Karina spends $$\boxed{30}$$ minutes bicycling.

[No calc]  Question  Hard

One of the two equations in a linear system is 2x + 2y = 2. The system has no solution. Which equation could be the other equation in the system?

A)3x − 3y = 3

B)3x + 3y = 3

C)2x − 2y = 2

D)2x + 2y = 3

D)2x + 2y = 3

For a linear system to have no solution, the equations must represent parallel lines that do not intersect. Parallel lines have the same slope but different y-intercepts.

The given equation is:
$2x + 2y = 2$

First, let’s rewrite this equation in slope-intercept form ($$y = mx + b$$) to find the slope:
$2y = -2x + 2$
$y = -x + 1$

The slope of this line is $$-1$$.

Now, we need to find another equation that has the same slope but a different y-intercept. Let’s examine each option:

A) $$3x – 3y = 3$$
Rewrite in slope-intercept form:
$-3y = -3x + 3$
$y = x – 1$
The slope is $$1$$, which is different from $$-1$$. This option is not parallel.

B) $$3x + 3y = 3$$
Rewrite in slope-intercept form:
$3y = -3x + 3$
$y = -x + 1$
The slope is $$-1$$, but the y-intercept is also $$1$$. This represents the same line, not a parallel one with a different y-intercept.

C) $$2x – 2y = 2$$
Rewrite in slope-intercept form:
$-2y = -2x + 2$
$y = x – 1$
The slope is $$1$$, which is different from $$-1$$. This option is not parallel.

D) $$2x + 2y = 3$$
Rewrite in slope-intercept form:
$2y = -2x + 3$
$y = -x + \frac{3}{2}$
The slope is $$-1$$, and the y-intercept is $$\frac{3}{2}$$. This line is parallel to the original line but has a different y-intercept.

The correct equation that could be the other equation in the system, resulting in no solution, is:D

[No – Calc]  Questions   Hard

4x+y=7
2x-7y=1

If (x, y) is the solution to the given system of equations, what is the value of x ?

Ans: 5/3, 1.66, 1.67

$\begin{cases} 4x + y = 7 \\ 2x – 7y = 1 \end{cases}$

We can solve this system using either the substitution method or the elimination method.

Multiplying the first equation by $$7$$ and the second equation by $$1$$ to eliminate $$y$$, we get:
$\begin{cases} 28x + 7y = 49 \\ 2x – 7y = 1 \end{cases}$

Adding the equations, we eliminate $$y$$:
$28x + 7y + 2x – 7y = 49 + 1$
$30x = 50$

Dividing both sides by $$30$$:
$x = \frac{50}{30}$
$x = \frac{5}{3}$

So, the value of $$x$$ is $$\frac{5}{3}$$.

[Calc]  Question  Hard

5x-y = 9
-60x+12y =- 108
How many solutions does the given system of equations have?
A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

Ans: D

To solve the system of equations:
\begin{align*} 5x – y &= 9 \\ -60x + 12y &= -108 \end{align*}

We can use the method of elimination.

First, let’s multiply the first equation by 12 and the second equation by -1 to eliminate $$y$$:

\begin{align*} 60x – 12y &= 108 \\ 60x – 12y &= 108 \end{align*}

Now, we can see that these equations are the same. This means they represent the same line. Therefore, the system of equations has infinitely many solutions.

So, the correct answer is option $$\mathbf{D}$$ – Infinitely many solutions.

[No- Calc]  Question  Hard

\begin{aligned} & 2 x+7 y=4 \\ & 8 x+4 y=12 \end{aligned}

If $$(x, y)$$ satisfies the given system of equations, what is the value of $$y$$ ?

Ans:1.6 or.166 or .167

Solve the second equation for $$x$$ or $$y$$. Let’s solve for $$x$$ in the second equation:
$8x + 4y = 12$
Divide by 4:
$2x + y = 3$

Use substitution or elimination. Subtract the first equation from this new equation:
$(2x + y) – (2x + 7y) = 3 – 4$
$y – 7y = -1$
$-6y = -1$
$y = \frac{1}{6}$

The value of $$y$$ is $$\frac{1}{6}$$.

[Calc]  Question   Hard

$$\begin{gathered} 2 x+6 y=2 \\ 2(2 x+y)=20 \end{gathered}$$

How many solutions does the given system of equations have?
A) Zero
B) Exactly one
C) Exactly two
D) Infinitely many

B

[Calc]  Question  Hard

$$\frac{1}{3}(x-k)=k x$$

In the given equation, $k$ is a constant. If the equation has no solution, what is the value of $k$ ?
A) -1
B) $-\frac{1}{3}$
C) 0
D) $\frac{1}{3}$

D

[Calc]  Question  Hard

$$\begin{array}{r} x+y=5 \\ 2 x=5 \end{array}$$

If $(x, y)$ is the solution to the given system of equations, what is the value of $y$ ?

$2.5,5 / 2$

Question

2$$x$$ + 3$$y$$ = 2
$$x$$ – 2$$y$$ = 8
The solution to the given system of equations is ($$x$$, $$y$$). What is the value of $$x$$ ?

4

Question

Data set X and data set Y are displayed by the two dot plots shown. Which of the following is(are) the same for both data sets?
1. The mean
2. The median

1. I only
2. II only
3. I and II
4. Neither I nor II

B

Question

The graph of the equation 4$$x$$+3$$y$$=$$q$$, where $$q$$ is a constant, is a line in the         $$xy$$- plane. What are the coordinates of the point at which the line crosses the $$x$$-axis?

1. $(\frac{q}{3},0)$
2. $(\frac{q}{4},0)$
3. $(\frac{3}{q},0)$
4. $(\frac{4}{q},0)$

B

Question

In the system of equations above, is a constant. If the system has no solution, what is the value of

1. -9
2. -6
3. 3
4. 6

B

Question

The solution to the given system of equations is . What is the value of

2

Question

3$$x$$+4$$y$$=35

2$$x$$+2$$y$$=15

The solution to the given system of equations is ($$x$$,$$y$$). What is the value of $$x$$+2$$y$$ ?

20

Questions

\begin{aligned} & \text { } x+y=2 \\ & x-y=3 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $x$ ?

Ans: $2.5,5 / 2$

Questions

\begin{aligned} & \text { 18. } 2 x+3 y=31 \\ & 3 x-y=30 \end{aligned}

If $(x, y)$ is the solution to the system of equations above, what is the value of $100 x+40 y$ ?

Ans: 1220

Questions

$3 m+2 p=24$
$m+p=10$

If $\left(m_1, p_1\right)$ is the solution to the system of equations above, what is the value of $p_1$ ?

Ans: 6

Questions

$3 x+2 y=16$
$6 x+2 y=28$

If the system of equation above has solution $(x, y)$, what is the value of $x+y$ ?

Ans: 6

Questions

$$2 k(x-2)=x-2$$

In the equation above, $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$ ?

Ans: . $5,1 / 2$

Questions

\begin{aligned} & 3 x+y=29 \\ & x=2 \end{aligned}

If $(x, y)$ is the solution to the given system of equations, what is the value of $y$ ?

Ans: 23

Question

$a(-3 x-1)+x=7 x-2$ The equation above has no solutions, and $a$ is a constant. What is the value of $a$ ?
A. $-\frac{7}{3}$
B. -2
C. 0
D. 2

Ans: B

Question

$\frac{1}{2} x=a$
$x+y=5 a$

In the system of equations above, $a$ is a constant such that $0<a<\frac{1}{3}$. If $(x, y)$ is a solution to the system of equations, what is one possible value of $y$ ?

Ans: $0<x<1$