Worked Examples: SAT Strategy
Example 1: Easy (Substitution Method)
Question: The solution to the given system of equations is $(x, y)$. What is the value of $y$?
x &= 8
x + 3y &= 26
SAT Strategy: When one variable is already isolated, use substitution immediately. Do not waste time rearranging this into standard form for your calculator.
- Substitute $x = 8$ directly into the second equation: $8 + 3y = 26$.
- Subtract 8 from both sides: $3y = 18$.
- Divide by 3: $y = 6$.
- Answer: $6$.
Example 2: Medium (Elimination & Manipulation)
Question: The solution to the given system of equations is $(x, y)$. What is the value of $30x$?
5y &= 10x + 11
-5y &= 5x - 21
SAT Strategy: Always look at the system
vertically before doing any work. Notice that the $y$ terms are already perfect opposites ($5y$ and $-5y$). Also, note that the question asks for $30x$, not just $x$.
- Add the two equations straight down: $(5y - 5y) = (10x + 5x) + (11 - 21)$.
- Simplify: $0 = 15x - 10$.
- Solve for $15x$: $15x = 10$.
- The question asks for $30x$. Simply multiply both sides by 2!
- $30x = 20$.
- Answer: $20$.
Example 3: Hard (No Solution with Constants)
Question: In the given system of equations, $h$ is a constant. If the system has
no solution, what is the value of $h$?
4x - 9y &= 9y + 5
hy &= 2 + 4x
SAT Strategy: "No solution" means the lines are parallel. Parallel lines in standard form ($Ax + By = C$) have proportional $x$ and $y$ coefficients ($\frac{A_1}{A_2} = \frac{B_1}{B_2}$), but different constants.
- Rearrange both equations into standard form $Ax + By = C$.
- Eq 1: $4x - 9y - 9y = 5 \implies 4x - 18y = 5$.
- Eq 2: $-4x + hy = 2$.
- Set up the ratio of the coefficients: $\frac{4}{-4} = \frac{-18}{h}$.
- Simplify and solve: $-1 = \frac{-18}{h} \implies -h = -18 \implies h = 18$.
- Answer: $18$.
Practice Questions
Level 1: Easy (Foundational SAT)
- $x = 6$
$x + 2y = 16$
The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in) - $y = 3x$
$2x + y = 20$
Which ordered pair $(x, y)$ is the solution to the given system of equations?
- A) $(4, 12)$
- B) $(5, 15)$
- C) $(10, 30)$
- D) $(12, 4)$
- $x + y = 12$
$x - y = 4$
The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in) - $2x + y = 10$
$4x + 2y = 20$
How many distinct solutions does the given system of equations have?
- A) Exactly one
- B) Exactly two
- C) Infinitely many
- D) Zero
- A family bought 2 adult tickets and 3 child tickets to a fair, paying a total of $\$35$. If an adult ticket costs $\$10$, what is the cost, in dollars, of one child ticket? (Grid-in)
Level 2: Medium (Standard SAT)
- $24x + y = 48$
$6x + y = 72$
The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in) - $y = -3x$
$4x + y = 15$
The solution to the given system of equations is $(x, y)$. What is the value of $x$?
- A) $1$
- B) $5$
- C) $15$
- D) $45$
- $\frac{x}{2} + \frac{y}{6} = 10$
$\frac{x}{2} + \frac{y}{3} = 14$
The solution to the given system of equations is $(x, y)$. What is the value of $y$? (Grid-in) - $2x + 3y = 18$
$2x - y = 6$
The solution to the given system of equations is $(x, y)$. What is the value of $x + y$?
- A) $3$
- B) $4.5$
- C) $7.5$
- D) $10.5$
- $y = 5x + 10$
One of the equations in a system of two linear equations is given. The system has no solution. Which equation could be the second equation in the system?
- A) $5x - y = 12$
- B) $-5x + y = 10$
- C) $5x + y = 12$
- D) $x - 5y = 10$
Level 3: Hard (Advanced SAT)
- $4x - 9y = 9y + 5$
$hy = 2 + 4x$
In the given system of equations, $h$ is a constant. If the system has no solution, what is the value of $h$? (Grid-in) - $x + 3y = 10$
$ax + by = 30$
In the given system of equations, $a$ and $b$ are constants. If the system has infinitely many solutions, what is the value of $a + b$?
- A) $4$
- B) $9$
- C) $10$
- D) $12$
- $3x - 4y = 10$
$6x - ky = 15$
In the given system of equations, $k$ is a constant. If the system has no solution, what is the value of $k$? (Grid-in) - $ax - 5y = 12$
$3x - by = 4$
In the given system of equations, $a$ and $b$ are constants. If the system has infinitely many solutions, what is the value of $a \times b$?
- A) $9$
- B) $15$
- C) $20$
- D) $27$
- A theater sold 120 tickets for a play. Adult tickets cost $\$15$ each and child tickets cost $\$8$ each. If the theater collected exactly $\$1,300$ in ticket sales, how many adult tickets were sold? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- $4x + 7y = 10$
$2x - y = -4$
The solution to the given system of equations is $(x, y)$. What is the value of $6x + 6y$?
- A) $-6$
- B) $0$
- C) $6$
- D) $14$
- $px + qy = 10$
$2x - 3y = 7$
In the given system of equations, $p$ and $q$ are constants. If the system has no solution, what is the value of $\frac{p}{q}$? (Grid-in) - $x + y = 8$
$y + z = 12$
$x + z = 14$
The solution to the given system of equations is $(x, y, z)$. What is the value of $x + y + z$? (Grid-in) - $y = mx + 4$
$y = 2x + b$
In the given system of equations, $m$ and $b$ are constants. The system has exactly one solution. Which of the following statements must be true?
- A) $m = 2$ and $b = 4$
- B) $m = 2$ and $b \neq 4$
- C) $m \neq 2$
- D) $b \neq 4$
- $\frac{x}{2} + \frac{y}{3} = 5$
$\frac{x}{4} - \frac{y}{6} = 1$
The solution to the given system of equations is $(x, y)$. What is the value of $x$? (Grid-in)
🔒 Answer Key & Brief Explanations
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