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

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

Digital SAT Math Practice Questions - Medium : Linear equations in two variables - New Syllabus

DSAT 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

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Question Medium

An artist paints and sells square tiles. The selling price \( P \), in dollars, of a painted tile is a linear function of the side length of the tile \( s \), in inches, as shown in the table below:

Side length, \( s \) (inches)Price, \( P \) (dollars)
38.00
618.00
928.00

Which of the following could define the relationship between \( s \) and \( P \)?

A) \( P = 3s + 10 \)

B) \( P = \frac{10}{3}s + 8 \)

C) \( P = \frac{10}{3}s – 2 \)

D) \( P = \frac{3}{10}s – \frac{1}{10} \)

▶️ Answer/Explanation
Solution

Ans: C

Slope: \( \frac{18 – 8}{6 – 3} = \frac{10}{3} \)

At \( s = 3 \), \( P = 8 \): \( 8 = \frac{10}{3} \cdot 3 + a \), \( a = -2 \)

Equation: \( P = \frac{10}{3}s – 2 \)

Question Medium

There are infinitely many solutions to which of the following equations?

I. \(4(x+1)+2=4x+6\)

II. \(4(x+1)+1=4x+7\)

A) I only

B) II only

C) I and II

D) Neither I nor II

▶️ Answer/Explanation
Solution

Ans: A

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

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

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:
\(4(x+1)+1=4x+7\)

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.

Question Medium

A city is planning to build a rock retaining wall, a monument, and a garden in a park. The table above shows four rock types that will be considered for use in the project. Also shown for each rock type is its weight per volume, in pounds per cubic foot \((lb/ft^3)\), and the cost per pound, in dollars. Only basalt, granite, and limestone will be used in the garden. The rocks in the garden will have a total weight of 1,000 pounds. If 330 pounds of granite is used, which of the following equations could show the relationship between the amounts, x and y, in \(ft^3\), for each of the other rock types used?

A) \(165x + 180y = 670\)

B) \(165x + 120y = 1,000\)

C) \(120x + 180y = 670\)

D) \(120x + 180y = 1,000\)

▶️ Answer/Explanation
Solution

Answer: C

Given: Granite weighs 330 pounds. Limestone weight is \(120x\), where \(x\) is its volume in \(ft^3\). Basalt weight is \(180y\), where \(y\) is its volume in \(ft^3\). Total weight of rocks in the garden is 1,000 pounds.

Equation: \(120x + 180y + 330 = 1,000\)

Subtract 330 from both sides:

\(120x + 180y = 670\)

Question Medium

Ball throw diagram

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

▶️ Answer/Explanation
Solution

Answer: C

Given equation: \( y = \frac{1}{2} x + 8 \)

Initial height \( x = 24 \) feet

Substitute \( x = 24 \):

\( y = \frac{1}{2} \times 24 + 8 \)

\( y = 12 + 8 \)

\( y = 20 \)

Maximum height after bouncing once is 20 feet

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