SAT MAth Practice questions – all topics
- Advanced Math Weightage: 35% Questions: 13-15
- Equivalent expressions
- Nonlinear equations in one variable and systems of equations in two variables
- Nonlinear functions
SAT MAth and English – full syllabus practice tests
Question Easy
The graph of a linear equation and the graph of a quadratic equation are shown. What is true about the point \((-1,4)\) ?
A) The point satisfies only the quadratic equation
B) The point satisfies only the linear equation.
C) The point satisfies both equations.
D) The point satisfies neither equation.
▶️Answer/Explanation
Ans:C
For the linear equation represented by the straight line: Let’s call this equation y = mx + b, where m is the slope and b is the y-intercept. The point (-1, 4) is on this line, so if I substitute x = -1 and y = 4, it should satisfy the equation. 4 = m(-1) + b Solving for b using the point’s coordinates gives b = 4 + m
For the quadratic equation represented by the parabola: Let’s call this y = ax^2 + bx + c, where a, b, c are constants. Substituting x = -1 and y = 4, we get:
$4 = a(-1)^2 + b(-1) + c$ $
4 = a + (-b) + c$
Therefore, the point (-1, 4) satisfies both the linear and quadratic equations shown.
Question Easy
A ball is thrown upward from a height of 3 feet above the ground. Assuming no air resistance, the function \(h\) defined by \(h(t)=-16 t^2+36 t+3\) models the ball’s height \(h(t)\), in feet, above the ground \(t\) seconds after it is thrown. Based on the model, what is the meaning of \(h(2)=11\) in this context?
A) The ball hits the ground 2 seconds after it is thrown.
B) The ball hits the ground 11 seconds after it is thrown.
C) The ball is 11 feet above the ground 2 seconds after it is thrown.
D) The ball is 2 feet above the ground 11 seconds after it is thrown.
▶️Answer/Explanation
Ans:C
The function \(h(t) = -16t^2 + 36t + 3\) models the height of the ball above the ground at time \(t\). Substituting \(t = 2\) into the function gives us:
\[
h(2) = -16(2)^2 + 36(2) + 3 = -64 + 72 + 3 = 11
\]
So, \(h(2) = 11\) means that the ball is 11 feet above the ground 2 seconds after it is thrown. Therefore, the correct answer is C) The ball is 11 feet above the ground 2 seconds after it is thrown.
Question Easy
\(x^2 + 2x + 1 = 4\)
What is the positive solution to the given equation?
▶️Answer/Explanation
Ans:1
To solve the equation \(x^2 + 2x + 1 = 4\) for the positive solution:
Rewrite the equation:
\[
x^2 + 2x + 1 – 4 = 0
\]
\[
x^2 + 2x – 3 = 0
\]
Factor the quadratic equation:
\[
(x + 3)(x – 1) = 0
\]
Solve for \(x\):
\[
x + 3 = 0 \quad \text{or} \quad x – 1 = 0
\]
\[
x = -3 \quad \text{or} \quad x = 1
\]
Since we are looking for the positive solution:
\[
x = 1
\]
Question Easy
\(x^{2}-x-12=0\)
What is the sum of the solutions to the given equation?
A. -7
B. -1
C. 1
D. 7
▶️Answer/Explanation
Ans: C
To find the solutions, we can factor the quadratic equation:
\[ x^2 – x – 12 = 0 \]
\[ x^2 – 4x +3x- 12 = 0 \]
\[ x(x – 4)+3(x – 4) = 0 \]
\[ (x-4)(x+3) = 0 \]
So, the solutions are \(x = 4\) and \(x = -3\).
The sum of the solutions is \(4 + (-3) = \boxed{\text{C) } -1}\).
Question Easy
An object is kicked from a platform. The equation \(h = -4.9t^2 + 15t + 5\) represents this situation, where h is the height of the object above ground, in meters, t seconds after it is kicked. Which number represents the height, in meters, of the platform ?
A) 0
B) 4.9
C) 5
D) 15
▶️Answer/Explanation
C) 5
The equation \(h = -4.9t^2 + 15t + 5\) represents the height of an object above the ground in meters, \(t\) seconds after it is kicked. We need to find the height of the platform from which the object is kicked.
1. Identify the height of the platform:
The height of the platform corresponds to the value of \(h\) when \(t = 0\).
2. Substitute \(t = 0\) into the equation:
\[
h = -4.9(0)^2 + 15(0) + 5
\]
\[
h = 5
\]
Thus, the height of the platform is:
\[ \boxed{5} \]
Question Easy
6x—9y>12
Which of the following inequalities is equivalent to the inequality above?
A.x—y>2
B.2x-3y>4
C.3x-2y>4
D.3y—2x>2
▶️Answer/Explanation
Ans B
Rationale
Choice B is correct. Both sides of the given inequality can be divided by 3 to yield 2x — 3y>4.
Choices A, C, and D are incorrect because they are not equivalent to (do not have the same solution set as) the given inequality. For example, the ordered pair (0, —1.5) is a solution to the given inequality, but it is not a solution to any of the inequalities in choices A, C, or D.
Question Easy
The graph of a system of a linear equation and a nonlinear equation is shown. What is the solution (x,y) to this system?
A.(0,0)
B.(0,2)
C.(2,4)
D. (4,0)
▶️Answer/Explanation
Ans C
Rationale
Choice C is correct. The solution to the system of two equations corresponds to the point where the graphs of the equations intersect. The graphs of the linear equation and the nonlinear equation shown intersect at the point (2, 4). Thus, the solution to the system is (2,4).
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question Easy
Which of the following is a solution to the equation \(2x^2-4=x^2\)?
A.1
B.2
C.3
D.4
▶️Answer/Explanation
Ans B
Rationale
Choice B is correct. Subtracting \(x^2\) from both sides of the given equation yields \(x^2 — 4 = 0\). Adding 4 to both sides of the equation gives \(x^2 = 4\). Taking the square root of both sides of the equation gives x = 2 or x = 2. Therefore, x = 2 is one solution to the original equation.
Alternative approach: Subtracting \(x^2\) from both sides of the given equation yields \(x^2 — 4 =0\). Factoring this equation gives \(x^2 —4 = (x+2)(x — 2) =0\), such that x = 2 or x = —2. Therefore, x = 2 is one solution to the original equation.
Choices A, C, and D are incorrect and may be the result of computation errors.
Question Easy
q—29r=s
The given equation relates the positive numbers q, r, and s. Which equation correctly expresses q in terms of s and r?
A. q=s-29r
B. q=s+29r
C.q=29rs
D.\(q=-\frac{s}{29r}\)
▶️Answer/Explanation
Ans B
Rationale
Choice B is correct. Adding 29r to each side of the given equation yields q = s + 29r. Therefore, the equation q = s+ 29r correctly expresses q in terms of r and s.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.
Question Easy
x+y=12
\(y=x^2\)
If (x,y) is a solution to the system of equations above, which of the following is a possible value of x?
A.0
B.1
C.2
D.3
▶️Answer/Explanation
Ans D
Rationale
Choice D is correct. Substituting \(x^2\) from the second equation for y in the first equation yields \(x+x^2= 12\). Subtracting 12 from both sides of this equation and rewriting the equation results in \(x^2+x—12= 0\). Factoring the left-hand side of this equation yields (x=3)(x+4)=0. Using the zero product property to solve for x, it follows that x —3 =0 and x +4 =0. Solving each equation for x yields x = 3 and x = —4, respectively. Thus, two possible values of x are 3 and —4. Of the choices given, 3 is the only possible value of x.
Choices A, B, and C are incorrect. Substituting 0 for x in the first equation gives 0+y=12,0r y=12 then, substituting 12 for y and 0 for x in the second equation gives \(12 = 0^2\), or 12 = (), which is false. Similarly, substituting 1 or 2 for x in the first equation yields y = 11 or y = 10, respectively; then, substituting 11 or 10 for y in the second equation yields a false statement.
Question Easy
\(P=\frac{W}{t}\)
The power P produced by a machine is represented by the equation above, where W is the work performed during an amount of time t. Which of the following correctly expresses W in terms of P and t ?
A.W=Pt
B.\(W=\frac{P}{t}\)
C.\(W=\frac{t}{P}\)
D.W=P+t
▶️Answer/Explanation
Ans A
Rationale
Choice A is correct. Multiplying both sides of the equation by t yields \(P.t=(\frac{W}{t}.t)\), or Pt = W, which expresses W in terms of P and t. This is equivalent to W = Pt.
Choices B, C, and D are incorrect. Each of the expressions given in these answer choices gives W in terms of P and t but doesn’t maintain the given relationship between W, P, and t. These expressions may result from performing different operations with t on each side of the equation. In choice B, W has been multiplied by t,and P has been divided by t. In choice C, W has been multiplied by t, and the quotient of P divided by t has been reciprocated. In choice D, W has been multiplied by t, and P has been added to t. However, in order to maintain the relationship between the variables in the given equation, the same operation must be performed with t on each side of the equation.
Question Easy
lx—2|=9
What is one possible solution to the given equation?
▶️Answer/Explanation
Ans 11, -7
Rationale
The correct answer is 11 or —7. By the definition of absolute value, if |x — 2| = 9,then x — 2 =9 or x— 2 = —9. Adding 2 to both sides of the equation x— 2 = 9 yields x= 11. Adding 2 to both sides of the equation x— 2 = —9 yields x = —7. Thus, the given equation, |x — 2| = 9, has two possible solutions, 11 and—7. Note that 11 and -7 are examples of ways to enter a correct answer.
Question Easy
\(x^2=64\)
Which of the following values of x satisfies the given equation?
A -8
B.4
C.32
D.128
▶️Answer/Explanation
Ans A
Rationale
Choice A is correct. Solving for x by taking the square root of both sides of the given equation yields x=8 or x = —8. Of the choices given, —8 satisfies the given equation.
Choice B is incorrect and may result from a calculation error when solving for x. Choice C is incorrect and may result from dividing 64 by 2 instead of taking the square root. Choice D is incorrect and may result from multiplying 64 by 2 instead of taking the square root.
Question Easy
The total revenue from sales of a product can be calculated using the formula T= PQ, where T is the total revenue, P is the price of the product, and Q is the quantity of the product sold. Which of the following equations gives the quantity of product sold in terms of P and T ?
A.\(Q=\frac{P}{T}\)
B.\(Q=\frac{T}{P}\)
C.Q=PT
D.Q=T-P
▶️Answer/Explanation
Ans B
Rationale
Choice B is correct. Solving the given equation for Q gives the quantity of the product sold in terms of Pand T. Dividing both sides of the given equation by P yields \(\frac{T}{P}=Q\) or \(Q=\frac{T}{P}\). Therefore, \(Q=\frac{T}{P}\) gives the quantity of product sold in terms of Pand T.
Choice A is incorrect and may result from an error when dividing both sides of the given equation by P. Choice C is incorrect and may result from multiplying, rather than dividing, both sides of the given equation by P. Choice D is incorrect and may result from subtracting P from both sides of the equation rather than dividing both sides by P.
Question Easy
b=42cf
The given equation relates the positive numbers b, c, and f. Which equation correctly expresses c in terms of b and f?
A.\(c=\frac{b}{42f}\)
B.\(c=\frac{b-42}{f}\)
C.c=42bf
D.c=42-b-f
▶️Answer/Explanation
Ans A
Rationale
Choice A is correct. It’s given that the equation b = 42cf relates the positive numbers b, c, and f. Dividing each side of the given equation by 42f yields \(\frac{b}{42f}=c\) or \(c=\frac{b}{42f}\). Thus, the equation \(c=\frac{b}{42f}\) correctly expresses c in terms of hand f.
Choice B is incorrect. This equation can be rewritten as b = cf + 42.
Choice C is incorrect. This equation can be rewritten as b = 42bf.
Choice D is incorrect. This equation can be rewritten as b =42 —c— f.
Question Easy
If \((x+ 5)^2 =4\), which of the following is a possible value of x ?
A.1
B.—1
C.-2
D.-3
▶️Answer/Explanation
Ans D
Rationale
Choice D is correct. If \((x+5)^2= 4\), then taking the square root of each side of the equation gives x +5=2 or x+5= —2. Solving these equations for x gives x = —3 or x = — 7. Of these, —3 is the only solution given as a choice.
Choice A is incorrect and may result from solving the equation x +5 = 4 and making a sign error. Choice B is incorrect and may result from solving the equation x +5 = 4. Choice C is incorrect and may result from finding a possible value of x 4+ 5.
Question Easy
\(\sqrt{x+4} =11\)
What value of x satisfies the equation above?
▶️Answer/Explanation
Ans Rationale
The correct answer is 117. Squaring both sides of the given equation gives \(x+4 =1 1^2\), or x+4=121. Subtracting 4 from both sides of this equation gives x =117.
Question Easy
\(x^2 = (22)(22)\) What is the positive solution to the given equation?
▶️Answer/Explanation
Ans 22
Rationale
The correct answer is 22. The given equation, \(x^2 = 2222\), is equivalent to \(x^2 = 22^{22}\). Taking the square root of each side of this equation yields \(x = \pm 22\). Thus, the positive solution to the given equation is 22.
Question Easy
p+34=q+r
The given equation relates the variables p, q, and . Which equation correctly expresses p in terms of q and r?
A.p=q+r+34
B.p=q+r—34
C.p=—q—r+34
D.p=—q—r—34
▶️Answer/Explanation
Ans B
Rationale
Choice B is correct. Subtracting 34 from each side of the given equation yields p = q + r – 34. Thus, the equation p = q +r- 34 correctly expresses p in terms of q and r.
Choice A is incorrect. This equation can be rewritten as p-34 = q +r.
Choice C is incorrect. This equation can be rewritten as p-34 = -q-r.
Choice D is incorrect. This equation can be rewritten as p + 34 = -q – r.