Worked Examples: SAT Strategy
Example 1: Easy (Isolating Variables)
Question: If $5x + 7 = 22$, what is the value of $5x$?
SAT Strategy: Read carefully! The SAT frequently asks for the value of an
expression rather than just $x$.
- You do not need to solve for $x$.
- Simply subtract 7 from both sides: $5x = 22 - 7$.
- Answer: $15$.
Example 2: Medium (Literal Equations)
Question: The equation $F = \frac{9}{5}(K - 273) + 32$ relates temperature in degrees Fahrenheit, $F$, to temperature in kelvins, $K$. Which equation correctly expresses $K$ in terms of $F$?
SAT Strategy: You must isolate $K$. Do this step-by-step using inverse operations.
- Subtract 32 from both sides: $F - 32 = \frac{9}{5}(K - 273)$.
- Multiply both sides by the reciprocal $\frac{5}{9}$: $\frac{5}{9}(F - 32) = K - 273$.
- Add 273 to both sides: $K = \frac{5}{9}(F - 32) + 273$.
Example 3: Hard (No Solution)
Question: $-3x + 21px = 84$. In the given equation, $p$ is a constant. If the equation has no solution, what is the value of $p$?
SAT Strategy: A linear equation has "no solution" when the $x$ terms on both sides cancel out, leaving a false statement (like $0 = 84$).
- Factor out the $x$ on the left side: $x(-3 + 21p) = 84$.
- For the $x$ to disappear, its coefficient must be equal to 0.
- Set the coefficient to zero: $-3 + 21p = 0$.
- Solve for $p$: $21p = 3 \implies p = \frac{3}{21} \implies p = \frac{1}{7}$.
- Answer: $1/7$.
Practice Questions
Level 1: Easy (Foundational SAT)
- $x + 16 = 30$. What value of $x$ is the solution to the given equation? (Grid-in)
- If $6n = 12$, what is the value of $n + 4$?
- Three more than eight times a number $x$ is equal to 83. Which equation represents this situation?
- A) $3x + 8 = 83$
- B) $8x + 3 = 83$
- C) $3(x + 8) = 83$
- D) $8(x + 3) = 83$
- $x - 53 = \frac{91}{2}$. Which equation has the same solution as the given equation?
- A) $2x - 53 = 91$
- B) $2x - 106 = 91$
- C) $x = \frac{38}{2}$
- D) $x = \frac{144}{2}$
- $7x(2 - 3) = 63$. Which equation has the same solution as the given equation?
- A) $-7x = 63$
- B) $7x - 3 = 63$
- C) $14x - 3 = 63$
- D) $-1x = 63$
- If $\frac{k}{5} = 15$, what is the value of $k - 10$?
Level 2: Medium (Standard SAT)
- If $4x - 28 = -24$, what is the value of $x - 7$?
- A) $-24$
- B) $-6$
- C) $-1$
- D) $1$
- $14x = 2w + 19y$. The given equation relates the distinct positive real numbers $w$, $x$, and $y$. Which equation correctly expresses $w$ in terms of $x$ and $y$?
- A) $w = 7x - \frac{19}{2}y$
- B) $w = 14x - 19y$
- C) $w = \frac{14x + 19y}{2}$
- D) $w = 7x - 19y$
- The equation $d = 30t$ gives the distance $d$, in feet, that a bus will travel $t$ seconds after passing a marker. How many feet from the marker will the bus be 2.5 seconds after passing it? (Grid-in)
- A veterinarian recommends that each day a certain rabbit should eat 25 calories per pound of the rabbit's weight, plus an additional 11 calories. Which equation represents this situation, where $c$ is the total number of calories if the rabbit's weight is $x$ pounds?
- A) $c = 25x$
- B) $c = 11x + 25$
- C) $c = 25x + 11$
- D) $c = 36x$
- $5.25x + 7.50y = 500$. The equation represents the total sales in dollars from selling two different-sized containers of fruit, where $x$ is the number of smaller containers sold and $y$ is the number of larger containers sold. According to the equation, what is the price, in dollars, of each smaller container? (Grid-in)
- The equation $V = P(1 + rt)$ gives the total value $V$ of an investment of $P$ dollars at a simple annual interest rate $r$ for $t$ years. Which equation correctly expresses $r$ in terms of $V$, $P$, and $t$?
- A) $r = \frac{V - 1}{Pt}$
- B) $r = \frac{V - P}{P}$
- C) $r = \frac{V - P}{Pt}$
- D) $r = \frac{V}{Pt} - P$
Level 3: Hard (Advanced SAT)
- $8x + 12 = kx + 12$. In the given equation, $k$ is a constant. If the equation has infinitely many solutions, what is the value of $k$? (Grid-in)
- $|2x - 5| = 11$. What is the positive solution to the given equation? (Grid-in)
- $cx - 64 = 48$. In the given equation, $c$ is a constant. If $x = 4$ is a solution to the equation, what is the value of $c$?
- The formula $A = \frac{1}{2}bh$ gives the area $A$ of a triangle with base $b$ and height $h$. Which equation correctly expresses $h$ in terms of $A$ and $b$?
- A) $h = \frac{A}{2b}$
- B) $h = \frac{2A}{b}$
- C) $h = 2Ab$
- D) $h = A - \frac{1}{2}b$
- $5(x - 3) + 2x = px - 15$. In the given equation, $p$ is a constant. If the equation has no solution, what is the value of $p$?
- $2(x - 3) + kx = 3x - 5$. In the given equation, $k$ is a constant. If the equation has no solution, what is the value of $k$? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- $3x + 15 = 3(x + p)$. In the given equation, $p$ is a constant. If the equation has infinitely many solutions, what is the value of $p$?
- A) 3
- B) 5
- C) 15
- D) It cannot be determined.
- $2(jx + 5) = 4x + k$. In the given equation, $j$ and $k$ are constants. If the equation has no solution, which of the following statements must be true?
- A) $j = 2$ and $k \neq 10$
- B) $j = 2$ and $k = 10$
- C) $j \neq 2$ and $k = 10$
- D) $j \neq 2$ and $k \neq 10$
- If $\frac{x}{y} = 4$ and $\frac{x}{ny} = 4$, what is the value of $n$? (Grid-in)
- $|x - 5| = c$. In the given equation, $c$ is a constant. If the equation has exactly one solution, what is the value of $c$? (Grid-in)
- $ax + b = cx + d$. In the given equation, $a$, $b$, $c$, and $d$ are constants. If $a = c$ and $b \neq d$, how many solutions does the equation have?
- A) Zero
- B) Exactly one
- C) Exactly two
- D) Infinitely many
- $(m+1)x + 2n = 3x - 6$. In the given equation, $m$ and $n$ are constants. If the equation has infinitely many solutions, what is the value of $m+n$? (Grid-in)
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