Heart of Algebra Previous Year Questions — SAT Math
17 past-year Heart of Algebra questions from SAT (Math).
Q3026 (2023)
What is the slope of the line perpendicular to the one shown below?
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- $\frac{3}{4}$
- $\frac{4}{3}$
- $-\frac{4}{3}$
- $-\frac{3}{4}$
### Related Formula
$$m_{\perp} = -\frac{1}{m}$$
where $m$ is the slope of the given line and $m_{\perp}$ is the slope of the line perpendicular to it.
### Core Logic
Identify two points on the line shown in the graph: $(0, -3)$ and $(4, 0)$.
Calculate the slope $m$ using the rise over run formula:
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-3)}{4 - 0} = \frac{3}{4}$$
The slope of any perpendicular line is the negative reciprocal of $\frac{3}{4}$.
### Step 1: Calculate Perpendicular Slope
$$m_{\perp} = -\frac{1}{3/4} = -\frac{4}{3}$$
Hence, choice C is correct.
### Pattern Recognition
Line graph $\rightarrow$ count rise/run to find $m$. Perpendicular slope is always the negative reciprocal flip.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3027 (2023)
Mark runs a tutoring service where he charges $\text{\$7}$ for each new client and $\text{\$20}$ per hour. His new client gets 4 hours of tutoring and pays Mark $\text{\$100}$. Which of the following statements is correct?
- Mark was overpaid
- Mark was underpaid
- You cannot determine if he was overpaid or underpaid based on the information given.
- Mark was paid the correct amount for his services.
### Related Formula
$$\text{Total Cost} = \text{Fixed Fee} + (\text{Hourly Rate} \times \text{Hours})$$
### Core Logic
Calculate Mark's required total fee for a new client getting 4 hours of tutoring:
$$\text{Fee} = \$7 + (\$20 \times 4) = \$7 + \$80 = \$87$$
Mark was paid $\$100$.
### Step 1: Compare Payment with Expected Fee
Since $\$100 > \$87$, Mark received $\$13$ more than his standard charge. Therefore, Mark was overpaid.
### Pattern Recognition
Word problem with fixed + variable cost $\rightarrow$ set up total equation, evaluate, compare to actual payout.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3028 (2023)
Which of the following lines has an undefined slope?
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- Red line
- Green line
- Blue line
- Orange line
### Related Formula
$$m = \frac{\Delta y}{\Delta x}$$
### Core Logic
A vertical line has a constant $x$-value, which means $\Delta x = 0$.
Since division by zero is mathematically undefined, vertical lines have an undefined slope.
Looking at the graph, the Blue Line is a vertical line at $x = 2$.
### Step 1: Conclusion
Therefore, the Blue line has an undefined slope.
### Pattern Recognition
Horizontal line $\rightarrow$ slope = 0. Vertical line $\rightarrow$ slope is undefined.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3029 (2023)
Megan and her friends all go to buy candy at the store. Mark pays \$15 for 3 chocolate bars, Megan pays \$20 for 4 chocolate bars, and Cindy pays \$100 for 20 chocolate bars. Which equation below represents the price per chocolate bar related to the total price of each order? Please note that $y$ represents total price and $x$ represents number of chocolate bars.
- $5y = x$
- $y = 5x$
- $y = x + 5$
- $15y = 3x$
### Related Formula
$y = kx$
where $k$ represents the unit price per item, $x$ represents quantity, and $y$ represents total cost.
### Core Logic
Determine the unit cost per chocolate bar from each order:
$$\text{Unit Price} = \frac{\$15}{3} = \frac{\$20}{4} = \frac{\$100}{20} = \$5$$
So $k = 5$.
### Step 1: Formulate Equation
Substitute $k = 5$ into $y = kx$:
$y = 5x$
Thus, option B is correct.
### Pattern Recognition
Proportional direct variation $\rightarrow$ total = unit rate $\times$ quantity.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3031 (2023)
What is the solution to the system of linear equations demonstrated in the graph?
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- $(0, 2)$
- $(0, 0)$
- $(2, 0)$
- $(2, 2)$
### Related Formula
The solution to a system of linear equations represented graphically is given by the point of intersection $(x, y)$ of the lines.
### Core Logic
Observe the point where the Red Line and Blue Line intersect on the coordinate plane.
They cross directly on the x-axis at $x = 2$ and $y = 0$.
### Step 1: Conclusion
Thus, the point of intersection is $(2, 0)$, corresponding to option C.
### Pattern Recognition
Graph of system $\rightarrow$ solution is simply the intersection point.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
