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
Question Medium
What value of \(x\) satisfies the equation \(\frac{1}{8} x+\frac{1}{4}=\frac{1}{2}\) ?
▶️Answer/Explanation
Ans: 2
To solve the equation \(\frac{1}{8} x + \frac{1}{4} = \frac{1}{2}\), follow these steps:
1. Subtract \(\frac{1}{4}\) from both sides:
\[
\frac{1}{8} x + \frac{1}{4} – \frac{1}{4} = \frac{1}{2} – \frac{1}{4}
\]
\[
\frac{1}{8} x = \frac{1}{2} – \frac{1}{4}
\]
2. Simplify the right side:
\[
\frac{1}{2} – \frac{1}{4} = \frac{2}{4} – \frac{1}{4} = \frac{1}{4}
\]
3. Now, solve for \(x\) by multiplying both sides by 8:
\[
\frac{1}{8} x = \frac{1}{4}
\]
\[
x = \frac{1}{4} \times 8 = 2
\]
So, the value of \(x\) is:
\[
\boxed{2}
\]
Question medium
How many solutions does the equation \(4(x-2)=-2(x+4)\) have?
A. Zero
B. Exactly one
C. Exactly two
D. Infinitely many
▶️Answer/Explanation
Ans:B
To find the number of solutions for the equation \(4(x-2) = -2(x+4)\), we’ll first simplify the equation:
\[4(x-2) = -2(x+4)\]
\[4x – 8 = -2x – 8\]
Add \(2x\) to both sides:
\[6x – 8 = -8\]
Add \(8\) to both sides:
\[6x = 0\]
Divide both sides by \(6\):
\[x = 0\]
The equation has exactly one solution \(x = 0\), so the correct answer is option B: Exactly one.
Question medium
5𝑟 = 3(𝑟 + 1)
What value of r is the solution to the given equation?
▶️Answer/Explanation
Ans: 3/2, 1.5
Let’s solve for \(r\) :
\[
\begin{gathered}
5 r=3(r+1) \\
5 r=3 r+3 \\
5 r-3 r=3 \\
2 r=3 \\
r=\frac{3}{2}
\end{gathered}
\]
So, the value of \(r\) that satisfies the equation is \(r=\frac{3}{2}\).
Question medium
\(\left | 15-x \right |-9=0\)
What are all the possible solutions to the given equation?
A. –6 and –24
B. –6 and 24
C. 6 and –24
D. 6 and 24
▶️Answer/Explanation
Ans: D
\[|15-x| = 9\]
Now, we can break this into two cases:
1. \(15 – x = 9\)
2. \(15 – x = -9\)
For case 1:
\[15 – x = 9\]
\[15 – 9 = x\]
\[x = 6\]
For case 2:
\[15 – x = -9\]
\[15 + 9 = x\]
\[x = 24\]
So, the possible solutions are \(x = 6\) and \(x = 24\).
Therefore, the correct answer is option D) 6 and 24.
Question Medium
If \(3\left(\frac{x}{5}+\frac{1}{2}\right)+1=10\), what is the value of \(\frac{x}{5}+\frac{1}{2}\) ?
A) 1
B) 3
C) 6
D) 12
▶️Answer/Explanation
Ans:B
\[
3\left(\frac{x}{5} + \frac{1}{2}\right) + 1 = 10
\]
1. Subtract 1 from both sides:
\[
3\left(\frac{x}{5} + \frac{1}{2}\right) = 9
\]
2. Divide both sides by 3:
\[
\frac{x}{5} + \frac{1}{2} = 3
\]
Question medium
\(3(2x-6)-11=4(x-3)+6\)
If x is the solution to the equation above, what is the value of \(x-3\)?
A \(\frac{23}{2}\)
B \(\frac{17}{2}\)
C \(\frac{15}{2}\)
D \(-\frac{15}{2}\)
▶️Answer/Explanation
Ans: B
In the input bar, type the left side of the equation as a function: y=3(2x−6)−11
In the next input bar, type the right side as another function: y=4(x−3)+6
Desmos will graph both functions as lines.
Question medium
\(2n+6=14\)
A tree had a height of 6 feet when it was planted. The equation above can be used to find how many years n it took the tree to reach a height of 14 feet. Which of the following is the best interpretation of the number 2 in this context?
A. The number of years it took the tree to double its height
B. The average number of feet that the tree grew per year
C. The height, in feet, of the tree when the tree was 1 year old
D. The average number of years it takes similar trees to grow 14 feet
▶️Answer/Explanation
Ans: B
Let’s tackle this step-by-step to figure out what the number 2 means in the equation \(2n+6=14\), where the tree starts at 6 feet and grows to 14 feet over n years.
Step 1: Solve the Equation
We’re given: \(2n+6=14\).
Subtract 6 from both sides:
\(2n=14−6\)
\(2n=8\).
Divide by 2:
\(n=\frac{8}{2} =4\)
So, it took 4 years (n=4) for the tree to reach 14 feet from its initial height of 6 feet.
Step 2: Break Down the Equation
The equation models the tree’s height:
6 is the height when planted (n=0).
2n is the additional height gained over nn years.
Total height = \(6+2n\).
At \(n=4: 2(4)+6=8+6=14\), which matches the problem.
Total growth: 14−6=8 feet in 4 years.
Growth per year: \( \frac{8}{4}=2\) feet per year.
The 2 multiplies n (years), so it seems tied to the growth rate. Let’s test it:
\(n=1: 2(1)+6=8 \) feet (grew 2 feet).
\(n=2: 2(2)+6=10\) feet (grew 4 feet).
\(n=3: 2(3)+6=12\) feet (grew 6 feet).
The pattern confirms: 2 feet of growth per year.
Question medium
\(2x+16=a(x+8)\)
In the given equation, \(a\) is a constant. If the equation has infinitely many solutions, what is the value of \(a\)?
▶️Answer/Explanation
Ans: 2
Here’s a simplified solution to find the value of \(a\) in the equation \(2x + 16 = a(x + 8)\) that results in infinitely many solutions:
For the equation to have infinitely many solutions, it must be an identity—true for all \(x\). This means the left side (\(2x + 16\)) and right side (\(a(x + 8)\)) must be equivalent.
– Expand the right side: \(a(x + 8) = ax + 8a\).
– Set it equal: \(2x + 16 = ax + 8a\).
– Match coefficients:
– \(x\) terms: \(2 = a\).
– Constants: \(16 = 8a\).
– Solve: \(16 = 8a\) → \(a = \frac{16}{8} = 2\).
– Check: If \(a = 2\), then \(2(x + 8) = 2x + 16\), and \(2x + 16 = 2x + 16\) is true for all \(x\).
Thus, \(a = 2\).
**Final Answer:** \(\boxed{2}\)
Question medium
\((b-2)x=8\)
In the given equation, b is a constant. If the equation has no solution, what is the value of b ?
A. 2
B. 4
C. 6
D. 10
▶️Answer/Explanation
Ans: A
Choice A is correct.
To find the value of \(b\) that makes \((b – 2)x = 8\) have no solution:
– For no solution, the equation must be a contradiction.
– If the coefficient of \(x\) is 0, and the constant isn’t:
\(b – 2 = 0\) → \(b = 2\).
– Then: \(0 \cdot x = 8\), or \(0 = 8\), which is false for all \(x\).
Final Answer:\(\boxed{2}\)
Question medium
\(a(3-x)-b=-1-2x\)
In the equation above, a and b are constants. If the equation has infinitely many solutions, what are the values of a and b ?
A. \(a=2\) and \(b=1\)
B. \(a=2\) and \(b=7\)
C. \(a=-2\) and \(b=5\)
D. \(a=-2\) and \(b=-5\)
▶️Answer/Explanation
Ans: B
To find the values of \(a\) and \(b\) that make the equation \(a(3 – x) – b = -1 – 2x\) have infinitely many solutions, it must be an identity—true for all \(x\). This means the left and right sides are equivalent after simplification.
– Expand the left side:
\(a(3 – x) – b = 3a – ax – b\).
– Compare with the right side:
\(3a – ax – b = -1 – 2x\).
– Equate coefficients:
– \(x\) term: \(-a = -2\) → \(a = 2\).
– Constant: \(3a – b = -1\).
Substitute \(a = 2\): \(3(2) – b = -1\) → \(6 – b = -1\) → \(b = 7\).
– Check:
Left: \(2(3 – x) – 7 = 6 – 2x – 7 = -1 – 2x\).
Right: \(-1 – 2x\).
They match for all \(x\).
Final Answer:\(a = 2\), \(b = 7\)
\(\boxed{a = 2, b = 7}\)
Question medium
An agricultural scientist studying the growth of corn plants recorded the height of a corn plant at the beginning of a study and the height of the plant each day for the next 12 days. The scientist found that the height of the plant increased by an average of 1.20 centimeters per day for the 12 days. If the height of the plant on the last day of the study was 36.8 centimeters, what was the height, in centimeters, of the corn plant at the beginning of the study?
▶️Answer/Explanation
Ans: 22.4
To find the height of the corn plant at the beginning of the study:
– The plant grew an average of 1.20 cm per day for 12 days.
– Total growth = \(1.20 \times 12 = 14.4\) cm.
– Final height = 36.8 cm.
– Initial height = Final height – Total growth = \(36.8 – 14.4 = 22.4\) cm.
**Final Answer:** \(\boxed{22.4}\)
Question medium
\(2(p+1)+8(p-1)=5p\)
What value of p is the solution of the equation above?
▶️Answer/Explanation
Ans:1.2
To solve \(2(p + 1) + 8(p – 1) = 5p\):
– Expand: \(2p + 2 + 8p – 8 = 5p\).
– Simplify: \(10p – 6 = 5p\).
– Subtract \(5p\): \(5p – 6 = 0\).
– Add 6: \(5p = 6\).
– Divide by 5: \(p = \frac{6}{5}\).
**Final Answer:** \(\boxed{\frac{6}{5}}\)
Question medium
If \(4x-\frac{1}{2}=-5\), what is the value of \(8x-1\)?
A. 2
B. \(-\frac{9}{8}\)
C. \(-\frac{5}{2}\)
D. \(-10\)
▶️Answer/Explanation
Ans: D
To find \(8x – 1\) given \(4x – \frac{1}{2} = -5\):
– Solve for \(4x\):
\(4x = -5 + \frac{1}{2} = -\frac{10}{2} + \frac{1}{2} = -\frac{9}{2}\).
– Double both sides (since \(8x = 2 \cdot 4x\)):
\(8x = 2 \cdot -\frac{9}{2} = -9\).
– Subtract 1:
\(8x – 1 = -9 – 1 = -10\).
**Final Answer:** \(\boxed{-10}\)
Question medium
Meganʼs regular wage at her job is p dollars per hour for the first 8 hours of work in a day plus 1.5 times her regular hourly wage for work in excess of 8 hours that day. On a given day, Megan worked for 10 hours, and her total earnings for that day were $137.50$. What is Meganʼs regular hourly wage?
A. $11.75$
B. $12.50$
C. $13.25$
D. $13.75$
▶️Answer/Explanation
Ans: B
Megan’s regular wage is \(p\) dollars per hour: 8 hours at \(p\), plus 2 hours (10 – 8) at \(1.5p\). Total earnings = $137.50.
– Total pay: \(8p + 2 \cdot 1.5p = 8p + 3p = 11p\).
– Equation: \(11p = 137.50\).
– Solve: \(p = \frac{137.50}{11} = 12.50\).
**Final Answer:** \(\boxed{12.50}\)
Question medium
The width of a rectangular dance floor is w feet. The length of the floor is 6 feet longer than its width. Which of the following expresses the perimeter, in feet, of the dance floor in terms of w ?
A.\(2w+6\)
B.\(4w+12\)
C.\(w^2+6\)
D. \(w^2+6w\)
▶️Answer/Explanation
Ans: B
– Width = \(w\) feet.
– Length = \(w + 6\) feet.
– Perimeter = \(2(\text{width} + \text{length}) = 2(w + w + 6) = 2(2w + 6) = 4w + 12\).
**Final Answer:** \(\boxed{4w + 12}\) (B)
Question medium
If \(2(x-5)+3(x-5)=10\), what is the value of \((x-5)\)?
A. 2
B. 5
C. 7
D. 12
▶️Answer/Explanation
Ans: A
Let \(y = x – 5\). Then:
\(2(x – 5) + 3(x – 5) = 2y + 3y = 5y\).
Given \(5y = 10\), solve:
\(y = \frac{10}{5} = 2\).
So, \(x – 5 = 2\).
**Final Answer:** \(\boxed{2}\)
Question medium
\(\frac{1}{4}(x+5)-\frac{1}{3}(x+5)=-7\)
What value of x is the solution to the given equation?
A. -12
B. -5
C. 79
D. 204
▶️Answer/Explanation
Ans: C
Factor \(x + 5\) from \(\frac{1}{4}(x + 5) – \frac{1}{3}(x + 5) = -7\):
\((x + 5) \left( \frac{1}{4} – \frac{1}{3} \right) = -7\).
\(\frac{1}{4} – \frac{1}{3} = \frac{3}{12} – \frac{4}{12} = -\frac{1}{12}\).
So, \((x + 5) \cdot -\frac{1}{12} = -7\).
Multiply by \(-12\): \(x + 5 = 84\).
Subtract 5: \(x = 79\).
**Final Answer:** \(\boxed{79}\)