Heart of Algebra Previous Year Questions — SAT Math
17 past-year Heart of Algebra questions from SAT (Math) — page 2.
Q3032 (2023)
The following graphs represent a system of linear equations. The lines shown are parallel. How many solutions will this system of equations have?
{{IMG1}}
- No solution
- One solution
- Infinite solutions
- Because we are not given the equations of the lines, it is impossible to know.
### Related Formula
Parallel lines have the same slope ($m_1 = m_2$) but different y-intercepts ($b_1 \neq b_2$).
### Core Logic
Parallel lines never cross or intersect at any point in the plane.
Since the solution of a system is defined as the set of all intersection points, parallel lines yield zero intersection points.
### Step 1: Conclusion
Therefore, the system has no solution.
### Pattern Recognition
Parallel lines $\rightarrow$ zero intersections $\rightarrow$ no solution.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3033 (2023)
Which of the following graphs demonstrates a system of linear equations with no solution?
i.
{{IMG1}}
ii.
{{IMG2}}
iii.
{{IMG3}}
iv.
{{IMG4}}
- i & ii
- ii & iv
- iii & iv
- ii & iii
### Related Formula
A system of linear equations has no solution if and only if the lines are parallel.
### Core Logic
Examine each graph for parallel lines:
- Graph i shows perpendicular lines (one vertical, one horizontal).
- Graph ii shows two horizontal parallel lines.
- Graph iii shows two vertical parallel lines.
- Graph iv shows intersecting lines.
### Step 1: Conclusion
Since graphs ii and iii depict parallel lines, both systems have no solution. Hence, option D is correct.
### Pattern Recognition
Identify parallel orientation in multi-graph sets $\rightarrow$ both represent no solution systems.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q3036 (2023)
A man buys $a$ apples and $g$ oranges, each costing 80 cents and 60 cents respectively. If the man buys less than 18 fruits and spends less than $12, which of the following is the correct inequality representing the above information?
- $a + g = 18$
$4a + 3g = 60$
- $a + g < 18$
$4a + 3g < 60$
- $a + g \le 18$
$4a + 3g \le 60$
- $a + g \le 18$
$4a + 3g = 60$
### Related Formula
$$\text{Total Cost (cents)} = 80a + 60g < 1200$$
since $\$12 = 1200$ cents.
### Core Logic
Fruit quantity condition: buys 'less than 18 fruits' $\rightarrow a + g < 18$.
Cost condition: spends 'less than $\$12$' $\rightarrow 80a + 60g < 1200$.
Simplify the cost inequality by dividing all terms by 20:
$$\frac{80a}{20} + \frac{60g}{20} < \frac{1200}{20} \implies 4a + 3g < 60$$
### Step 1: Conclusion
The system of inequalities is $a + g < 18$ and $4a + 3g < 60$ (Option B).
### Pattern Recognition
'Less than' strictly means $<$, not $\leq$. Convert dollars to cents before simplifying.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q5 (2023)
Samantha bought M&Ms from the store. She is told that the amount of sweet M&Ms and the amount of bitter M&Ms is represented by the following system of inequalities, where $s$ is the number of sweet M&Ms and $b$ is the number of bitter M&Ms:
$s + b \geq 25$
$5s - 10b < 2$
Which of the following solutions, in the format $(s, b)$ could be a possible solution for the number of M&Ms in Samantha's bag?
- $(6, 10)$
- $(2, 30)$
- $(1, 1)$
- $(50, 10)$
### Related Formula
For a coordinate pair $(s, b)$ to be a valid solution to a system of inequalities, it must satisfy all given inequalities simultaneously.
### Core Logic
Test the ordered pair $(2, 30)$ where $s = 2$ and $b = 30$:
First inequality:
$$2 + 30 = 32 \geq 25 \quad \text{(True)}$$
Second inequality:
$$5(2) - 10(30) = 10 - 300 = -290 < 2 \quad \text{(True)}$$
### Step 1: Conclusion
Since both statements evaluate to true for $(2, 30)$, option B is a valid solution.
### Pattern Recognition
System of inequalities question $\rightarrow$ plug and check given options to find which makes both true.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
Q9 (2023)
If $a + 2b + 3c = 24$ and $3a + 2b + c = 36$, what is the value of $(a + b + c)$?
- 8
- 12
- 15
- 60
### Related Formula
For symmetric linear systems, adding equations directly often isolates the required expression $(a + b + c)$ without solving for individual variables.
### Core Logic
Given equations:
$$a + 2b + 3c = 24 \quad \dots (1)$$
$$3a + 2b + c = 36 \quad \dots (2)$$
Add (1) and (2) together:
$$(a + 3a) + (2b + 2b) + (3c + c) = 24 + 36$$
$$4a + 4b + 4c = 60$$
### Step 1: Solve for (a + b + c)
Factor out 4:
$$4(a + b + c) = 60$$
Divide by 4:
$a + b + c = 15$
Thus, the correct answer is C.
### Pattern Recognition
Symmetric coefficients $\rightarrow$ add equations directly to avoid tedious multi-variable elimination.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
