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
One of the two linear equations in a system is \(3 x+4 y=8\). The system has exactly one solution.
Which of the following could be the other equation in the system?
A) \(\frac{3}{2} x+2 y=4\)
B) \(3 x+4 y=4\)
C) \(4 x+3 y=8\)
D) \(6 x+8 y=16\)
▶️Answer/Explanation
Ans: C
To determine which equation would allow the system to have exactly one solution, the slopes of the two lines must be different. This ensures that the lines intersect at exactly one point.
The given equation is:
\[ 3x + 4y = 8 \]
First, we rewrite this equation in slope-intercept form (\(y = mx + b\)) to find its slope:
\[ 3x + 4y = 8 \]
\[ 4y = -3x + 8 \]
\[ y = -\frac{3}{4}x + 2 \]
The slope of the given equation is \(-\frac{3}{4}\).
Now, let’s check the slopes of the other options:
Option A: \(\frac{3}{2}x + 2y = 4\)
\[ \frac{3}{2}x + 2y = 4 \]
\[ 2y = -\frac{3}{2}x + 4 \]
\[ y = -\frac{3}{4}x + 2 \]
The slope here is also \(-\frac{3}{4}\), the same as the given equation. This means the lines are parallel and would either be identical (infinitely many solutions) or parallel (no solutions), not intersecting at one point.
Option B: \(3x + 4y = 4\)
This is the same equation as the given one but with a different constant term, which makes it a parallel line. Therefore, the lines are parallel and will not intersect at one point.
Option C: \(4x + 3y = 8\)
\[ 4x + 3y = 8 \]
\[ 3y = -4x + 8 \]
\[ y = -\frac{4}{3}x + \frac{8}{3} \]
The slope here is \(-\frac{4}{3}\), which is different from \(-\frac{3}{4}\). This means the lines have different slopes and will intersect at exactly one point.
Option D: \(6x + 8y = 16\)
This is a multiple of the given equation:
\[ 6x + 8y = 16 \]
\[ \frac{6x + 8y}{2} = \frac{16}{2} \]
\[ 3x + 4y = 8 \]
This is the same equation as the given one, just scaled by a factor of 2. Therefore, they are identical and would either have infinitely many solutions or none if they were parallel lines, not intersecting at one point.
So, the correct answer is:
\[
\boxed{C}
\]
Questions medium
$
h(x)=3 x+3
$
Which inequality represents all values of \(x\) for which the graph of \(y=h(x)\) in the \(x y\)-plane is above the \(x\) axis?
A. \(x<3\)
B. \(x<-1\)
C. \(x>-1\)
D. \(x>3\)
▶️Answer/Explanation
Ans:C
The function \(h(x) = 3x + 3\) represents a linear function with a slope of \(3\) and a \(y\)-intercept of \(3\). The graph of \(h(x)\) will be above the \(x\)-axis for all values of \(x\) greater than the \(x\)-intercept, which can be found by setting \(h(x) = 0\):
\[
3x + 3 = 0
\]
\[
3x = -3
\]
\[
x = -1
\]
Therefore, all values of \(x\) greater than \(-1\) will make the graph of \(h(x)\) above the \(x\)-axis. So, the correct inequality is \(x > -1\).
Question medium
\(y\leq =2x+3\)
\(y\geq 0.5x-6\)
In which graph does the shaded region represent all solutions to the given system of inequalities?
▶️Answer/Explanation
Ans: B
Given inequalities:
1. \(y \leq 2 x+3\)
2. \(y \geq 0.5 x-6\)
1. \(y \leq 2 x+3\) :
- The boundary line is \(y=2 x+3\).
- Since it is \(y \leq\), the region below this line will be shaded.
2. \(y \geq 0.5 x-6\) :
- The boundary line is \(y=0.5 x-6\).
- Since it is \(y \geq\), the region above this line will be shaded.
The correct solution will be the intersection of these two shaded regions.
Graph B:
- The region below the line \(y=2 x+3\) is shaded. (Correct)
- The region above the line \(y=0.5 x-6\) is shaded. (Correct)
- The intersection of these two regions is correctly shaded. (Correct)
Question Medium
Sanjay works as a teacher’s assistant for \(\$ 20\) per hour and tutors privately for \(\$ 25\) per hour. Last week, he made at least \(\$ 100\) working \(x\) hours as a teacher’s assistant and \(y\) hours as a private tutor. Which of the following inequalities models this situation?
A) \(4 x+5 y \geq 25\)
B) \(4 x+5 y \geq 20\)
C) \(5 x+4 y \geq 25\)
D) \(5 x+4 y \geq 20\)
▶️Answer/Explanation
Ans:B
Sanjay earns \( \$20 \) per hour as a teacher’s assistant and \( \$25 \) per hour as a private tutor. He worked \( x \) hours as a teacher’s assistant and \( y \) hours as a private tutor and made at least \( \$100 \).
To model this situation, we set up the following inequality:
total earnings from working \( x \) hours as a teacher’s assistant:
\[
20x
\]
total earnings from working \( y \) hours as a private tutor:
\[
25y
\]
3. The total earnings must be at least \( \$100 \):
\[
20x + 25y \geq 100
\]
Simplify this inequality by dividing everything by 5:
\[
4x + 5y \geq 20
\]
Question medium
According to a 2008 study, there were five known subspecies of tigers, including the Amur and Bengal, living in the wild. Scientists estimated that there were a total of 4,000 tigers in the wild. Of these, 450 were Amur tigers and x were Bengal tigers. Which inequality represents all possible numbers of tigers in the wild in 2008 that belonged to the Bengal tiger subspecies?
A. 5𝑥 ≥ 4000
B. \(\frac{x}{5}\geq 4000\)
C. 1 ≤ 𝑥 ≤ 3547
D. 3547 < 𝑥 ≤ 4000
▶️Answer/Explanation
Ans: C
To solve this problem, we need to find the range of possible values for the number of Bengal tigers (x) based on the given information.
- There were a total of 4,000 tigers in the wild.
- Out of these 4,000 tigers, 450 were Amur tigers.
- There were 5 known subspecies of tigers in the wild.
Since we know the total number of tigers and the number of Amur tigers, we can find the number of tigers belonging to the remaining 4 subspecies (including the Bengal tigers) by subtracting the number of Amur tigers from the total.
Number of tigers belonging to the remaining 4 subspecies = 4,000 – 450 = 3550
Since the Bengal tigers are one of the 4 remaining subspecies, the number of Bengal tigers (x) must be less than or equal to 3,550.
Also, the number of Bengal tigers cannot be negative or zero, as it is stated that they were one of the known subspecies living in the wild.
Therefore, the inequality that represents all possible numbers of tigers in the wild in 2008 that belonged to the Bengal tiger subspecies is:
$1 ≤ x ≤ 3550$
Among the given options, the correct answer is $C. 1 ≤ x ≤ 3547$.