Worked Examples: SAT Strategy
Example 1: Easy (The Linear Combination Shortcut)
Question: If $5x + 3y = 31$ and $5x - 3y = 19$, what is the value of $10x$?
SAT Strategy: Look at the question carefully! It does not ask for $x$, it asks for $10x$. Do not solve for $x$ and $y$ individually if you can jump straight to the answer.
- Notice that adding the two equations together perfectly eliminates the $y$ terms and sums to exactly $10x$.
- $(5x + 5x) + (3y - 3y) = 31 + 19$.
- $10x = 50$.
- Answer: $50$.
Example 2: Medium (No Solution / Parallel Lines)
Question: The system of equations $12x - 4y = 8$ and $rx - y = 5$ has no solution. What is the value of the constant $r$?
SAT Strategy: "No solution" means the lines are parallel. If equations are in standard form ($Ax + By = C$ and $Dx + Ey = F$), the coefficients of $x$ and $y$ must be perfectly proportional ($\frac{A}{D} = \frac{B}{E}$), but the constants must not be ($\neq \frac{C}{F}$).
- Set up the proportion for the $x$ and $y$ coefficients: $\frac{12}{r} = \frac{-4}{-1}$.
- Simplify the right side: $\frac{-4}{-1} = 4$.
- Solve for $r$: $\frac{12}{r} = 4 \implies 12 = 4r \implies r = 3$.
- (Check the constants: $\frac{8}{5} \neq 4$, so they are definitely distinct parallel lines).
- Answer: $3$.
Example 3: Hard (Infinitely Many Solutions)
Question: The system $ax + by = 15$ and $2x + 5y = 3$ has infinitely many solutions, where $a$ and $b$ are constants. What is the value of $a + b$?
SAT Strategy: "Infinitely many solutions" means the two equations represent the exact same line. One equation is just a scalar multiple of the other.
- Compare the constants: The second equation equals $3$, and the first equals $15$.
- To make the second equation match the first, multiply the entire second equation by $5$.
- $5 \times (2x + 5y = 3) \implies 10x + 25y = 15$.
- Now match the coefficients: $a = 10$ and $b = 25$.
- Find the requested sum: $a + b = 10 + 25 = 35$.
- Answer: $35$.
Practice Questions
Level 1: Easy (Foundational SAT)
- $y = 4x$
$2x + y = 24$
The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in) - $x + y = 18$
$x - y = 6$
What is the value of $x$?
- A) $6$
- B) $12$
- C) $18$
- D) $24$
- $3x - 2y = 10$
$5x + 2y = 22$
The solution to the given system is $(x, y)$. What is the value of $8x$? (Grid-in) - A system of two linear equations has no solution. Which of the following must be true about the graphs of the two equations in the $xy$-plane?
- A) They are perpendicular lines.
- B) They are intersecting lines with different slopes.
- C) They are parallel lines.
- D) They are the exact same line.
- $x = 8$
$x + 3y = 26$
The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in)
Level 2: Medium (Standard SAT)
- $2x + 3y = 15$
$x + 2y = 10$
What is the value of $x + y$?
- A) $5$
- B) $10$
- C) $15$
- D) $25$
- $6x + 8y = 10$
$3x + 4y = c$
If the given system of equations has infinitely many solutions, what is the value of $c$? (Grid-in) - $y = 5x + 4$
$y = mx - 2$
If the given system of equations has no solution, what is the value of $m$?
- A) $-2$
- B) $4$
- C) $5$
- D) $7$
- $\frac{x}{2} + \frac{y}{3} = 5$
$x + y = 12$
The solution to the given system is $(x, y)$. What is the value of $x$? (Grid-in) - $4x - y = 9$
$y = x - 3$
The solution to the given system is $(x, y)$. What is the value of $y$?
- A) $-1$
- B) $1$
- C) $2$
- D) $5$
Level 3: Hard (Advanced SAT)
- $12x + 8y = 40$
$3x + cy = 10$
In the given system of equations, $c$ is a constant. If the system has infinitely many solutions, what is the value of $c$? (Grid-in) - $kx - 4y = 7$
$3x + 2y = 5$
In the given system of equations, $k$ is a constant. If the system has no solution, what is the value of $k$? (Grid-in) - $\frac{x}{y} = 5$
$\frac{x}{ny} = 20$
The given system of equations relates the positive variables $x$, $y$, and $n$. What is the value of $n$?
- A) $0.25$
- B) $4$
- C) $15$
- D) $100$
- $24x + y = 48$
$6x + y = 72$
The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in) - $ax + by = 12$
$2x + 3y = 4$
The given system of equations has infinitely many solutions, where $a$ and $b$ are constants. What is the value of $a + b$?
- A) $5$
- B) $15$
- C) $20$
- D) $36$
Level 4: Very Hard (Slightly beyond SAT)
- $y = 6x + 18$
$cx - y = -22$
In the given system of equations, $c$ is a constant. If the system has no solution, what is the value of $c$? (Grid-in) - $px + qy = 10$
$3x + 4y = 8$
If the system of equations has no solution, which of the following correctly expresses $p$ in terms of $q$?
- A) $p = \frac{3q}{4}$
- B) $p = \frac{4q}{3}$
- C) $p = \frac{10q}{8}$
- D) $p = \frac{8q}{10}$
- $48x - 64y = 48 + 24y$
$rx - 11y = 6$
In the given system of equations, $r$ is a constant. If the system has infinitely many solutions, what is the value of $r$? (Grid-in) - $3x - 5y = 14$
$ax + by = c$
The graph of the second equation in the system passes through the points $(0, 4)$ and $(6, 0)$. If the system has exactly one solution, which of the following could be the value of $a$?
- A) $2$
- B) $3$
- C) $-1.8$
- D) Any real number except $-1.8$
- At a theater, the price of an adult ticket is $\$10$ and the price of a child ticket is $\$5$. On a certain night, $50$ tickets were sold for a total revenue of $\$350$. How many adult tickets were sold? (Grid-in)
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