Home / Digital SAT Math Practice Questions – Medium : Linear equations in two variables

# Digital SAT Math Practice Questions – Medium : 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

There are infinitely many solutions to which of the following equations?
I. $$4(x+1)+2=4 x+6$$
II. $$4(x+1)+1=4 x+7$$
A) I only
B) II only
C) I and II
D) Neither I nor II

Ans:A

We need to determine which of the given equations has infinitely many solutions.

Equation I:
$4(x+1)+2=4x+6$

First, simplify the left side:
$4x + 4 + 2 = 4x + 6 \\ 4x + 6 = 4x + 6$

This equation is always true for any value of $$x$$, which means there are infinitely many solutions for this equation.

Equation II:
$4(x+1)+1=4x+7$

First, simplify the left side:
$4x + 4 + 1 = 4x + 7 \\ 4x + 5 = 4x + 7$

This equation simplifies to:
$4x + 5 = 4x + 7 \implies 5 = 7$

This is a contradiction, so there are no solutions for this equation.

$\boxed{\text{I only}}$

[No calc]  Question   medium

How many solutions does the equation
3x − 8 = x + 2(x − 4) have?

A)  Zero

B) Exactly one

C) Exactly two

D) Infinitely many

D)  Infinitely many

To determine how many solutions the equation $$3x – 8 = x + 2(x – 4)$$ has, follow these steps:

Distribute the terms on the right-hand side:
$3x – 8 = x + 2x – 8$
$3x – 8 = 3x – 8$

Simplify the equation:
$3x – 8 = 3x – 8$

This is an identity, meaning it holds true for all $$x$$. Therefore, the equation has infinitely many solutions.

[Calc]  Question  medium

x(x − 12) − 12(x − 12) = 0

How many distinct real solutions does the given equation have?

A) Zero

B) Exactly one

C) Exactly two

D) Infinitely many

B) Exactly one

To find the number of distinct real solutions for the equation $$x(x – 12) – 12(x – 12) = 0$$, let’s first simplify it:

Distribute and combine like terms:
$x(x – 12) – 12(x – 12) = 0$
$x^2 – 12x – 12x + 144 = 0$
$x^2 – 24x + 144 = 0$

Now, we can see that this is a quadratic equation in the form $$ax^2 + bx + c = 0$$, where $$a = 1$$, $$b = -24$$, and $$c = 144$$.

We can use the discriminant ($$b^2 – 4ac$$) to determine the number of distinct real solutions:
$\text{Discriminant} = (-24)^2 – 4(1)(144) = 576 – 576 = 0$

Since the discriminant is equal to 0, the quadratic equation has exactly one real solution.

[Calc]  Question   Medium

The function f is defined by f(x) = -2x + 8. The x-intercept of the graph of y = f(x) in the xy-plane is (x, 0). What is the value of x ?

4

To find the $$x$$-intercept of the graph of $$y = f(x)$$ where $$f(x) = -2x + 8$$, we set $$y = 0$$ and solve for $$x$$.

1. Set $$y = 0$$:
$0 = -2x + 8$

2. Solve for $$x$$:
$2x = 8$
$x = 4$

[Calc]  Question   Medium

When a ball is thrown straight down with an initial speed of’32 feet per second, the ball hits the ground and bounces up. The equation $$y = \frac{1}{2} x + 8$$ represents the relationship between the ball’s initial height x, in feet, and the maximum height y, in feet, that the ball reaches after bouncing once.

If the initial height is 24 feet, what is the maximum height, in feet, the ball reaches after bouncing once ?
A) 12
B) 16
C) 20
D) 32

C) 20

Given the equation $$y = \frac{1}{2} x + 8$$ that represents the relationship between the ball’s initial height $$x$$, in feet, and the maximum height $$y$$, in feet, that the ball reaches after bouncing once, we need to find the maximum height when the initial height is 24 feet.

1. Substitute $$x = 24$$ into the equation:
$y = \frac{1}{2} \times 24 + 8$
$y = 12 + 8$
$y = 20$

Thus, the maximum height the ball reaches after bouncing once is:
$\boxed{20}$

[Calc]  Question Medium

The function $$f$$ is defined by $$f(x)=2 x+6$$. What is the graph of $$y=f(x)$$ ?

A

Y-intercept
The $$y$$-intercept occurs where the graph of the function crosses the $$y$$-axis. This happens when $$x=0$$.
$f(0)=2(0)+6=6$

Thus, the $$y$$-intercept is at $$(0,6)$$.
X-intercept
The $$\mathrm{x}$$-intercept occurs where the graph of the function crosses the $$\mathrm{x}$$-axis. This happens when $$f(x)=0$$.
$2 x+6=0$

Solve for $$x$$ :
\begin{aligned} & 2 x=-6 \\ & x=-3 \end{aligned}

Thus, the $$x$$-intercept is at $$(-3,0)$$.

[Calc]  Question  Medium

Alia sells nature photos and videos. She receives a fixed amount for each photo she sells and a different fixed amount for each video she sells. The graph models the possible numbers of photos p, and videos, v. Alia can sell to receive 300. Which equation represents this situation ?

A) pv=300

B) 20p + 50v = 300

C) 50p + 20v = 300

D) 15p + 6v= 300

B) 20p + 50v = 300

When Alia sells only 6 videos

6 videos=300

v= 50

When Alia sells only 15 Photos

15 Photos=300

P= 20

Checking the each option by putting value of v and p The correct match is option B

[Calc]  Question   Medium

When a ball is thrown straight down with an initial speed of’32 feet per second, the ball hits the ground and bounces up. The equation $$y = \frac{1}{2} x + 8$$ represents the relationship between the ball’s initial height x, in feet, and the maximum height y, in feet, that the ball reaches after bouncing once.

If the maximu:tn height the ball reaches after bouncing once is also x feet, what is the value of x ?
A) 4
B) $$\frac{16}{3}$$
C) 8
D) 16

D) 16

Given that the maximum height the ball reaches after bouncing once is also $$x$$ feet, we can use the equation $$y = \frac{1}{2} x + 8$$ to find the value of $$x$$.

1. Set $$y = x$$ in the equation:
$x = \frac{1}{2} x + 8$

2. Solve for $$x$$:
$x – \frac{1}{2} x = 8$
$\frac{1}{2} x = 8$
$x = 16$

Thus, the value of $$x$$ is:
$\boxed{16}$

[Calc]  Question   Medium

Line $p$ is defined by $2 y+4 x=9$. Line $r$ is perpendicular to line $p$ in the $x y$-plane. What is the slope of line $r$ ?

$.5,1 / 2$

[Calc]  Question  Medium

In the $x y$-plane, what is the $y$-coordinate of the $y$-intercept of the graph of the equation $y=\frac{3 x-12}{x+2}$ ?
A) -6
B) -2
C) 3
D) 4

A

[Calc]  Question  Medium

A sports store had 60 backpacks in stock, some with wheels and some without wheels, before a new shipment of backpacks arrived. The number of wheeled backpacks in the new shipment was twice the number of wheeled backpacks already in stock, and the number of backpacks without wheels in the new shipment was five times the number of backpacks without wheels already in stock. After the new shipment arrived, there were 330 backpacks in the store. Before the shipment, there were $x$ wheeled backpacks and $y$ backpacks without wheels. Which of the following equations can be used with $x+y=60$ to solve for $x$ and $y$ ?
A) $2 x+5 y=330$
B) $2 x+5 y=270$
C) $5 x+2 y=270$
D) $5 x+2 y=330$

B

Question

Henri buys 2 boxes of blue pens and some boxes of black pens, Each box contains 10 pens, and Henri buys a total of 50 pens. The equation 10($$x$$ + 2) = 50 represents this situation. Which of the following is the best interpretation of the expression $$x$$ + 2 in this context? 1.2

1. The number of blue pens that Henri buys
2. The number of boxes of pens that Henri buys
3. The number of boxes of blue pens that Henri buys
4. The number of boxes of black pens that Henri buys

B

Question

For the linear function $$f$$,  $$f$$(6) = 4, and the graph of $$y$$ = $$f$$($$x$$) in the $$xy$$-plane has a slope of $$\frac{1}{2}$$. Which equation defines $$f$$? 2.15

1. $$f(x)=\frac{1}{2}x+1$$
2. $$f(x)=\frac{1}{2}x+2$$
3. $$f(x)=\frac{1}{2}x+4$$
4. $$f(x)=\frac{1}{2}x+10$$

A

Question

A person used a total of 265 kilocalories (kcal) while walking and running on a treadmill.
Running at a constant rate required 11.5 kcal per minute, and walking at a constant rate required 3.5 kcal per minute. The relationship between the number of minutes running, , and the number of minutes walking, , is given by the equation shown. If this person ran for 20 minutes, how many minutes did this person walk?

1. 35
2. 29
3. 17
4. 10

D

Questions

In the $x y$-plane, the $y$-coordinate of the $y$-intercept of the graph of the function $f$ is $c$. Which of the following must be equal to $c$ ?
A. $f(0)$
B. $f(1)$
C. $f(2)$
D. $f(3)$

Ans: A

Question

The table shown gives some values of $$x$$ and the corresponding values of $$f(x)$$, where $$f$$ is a linear function. If $$y = f (x)$$ is graphed in the $$xy$$-plane, what is the $$y$$-coordinate of the $$y$$-intercept of the graph?

1. 1
2. 0.5
3. 0
4. -1

Ans: C

Questions

For gym class, Shayla completed a 4-mile walking and running exercise. She ran for $7 t$ miles and she walked for $3\left(\frac{13}{15}-t\right)$ miles, where $t$ is the total amount of time, in hours, Shayla spent running. The equation $7 t+3\left(\frac{13}{15}-t\right)=4$ models this situation.
What is the value of $t$ in the equation that models this situation?
A. $\frac{7}{50}$
B. $\frac{7}{20}$
C. $\frac{31}{60}$
D. $\frac{13}{15}$

Ans: B

Question

The function $f$ is linear, $f(2)=17$, and $f(8)=19$. If $f(x)=m x+b$, where $m$ and $b$ are constants, what is the value of $b$ ?
A. 11
B. 13
C. $\frac{49}{3}$
D. $\frac{55}{3}$

Ans: C

Questions

$y=-2$
$y+11=x^2$

If $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ are solutions to the system of equations above, what are the values of $x_1$ and $x_2$ ? 2.12
A. $-\sqrt{13}$ and $\sqrt{13}$
B. $-\sqrt{11}$ and $\sqrt{11}$
C. -2 and 2
D. -3 and 3

Ans: D

Questions

In the $x y$-plane, the point $(2,6)$ lies on the graph of $y=\frac{k}{x}$, where $k$ is a constant. Which of the following points must also lie on the graph?
A. $(1,3)$
B. $(1,4)$
C. $(3,3)$
D. $(3,4)$