The following graphs represent a system of linear equations. The lines shown are parallel. How many solutions will this system of equations have?
A graph showing two parallel lines that never intersect.
A.No solution
B.One solution
C.Infinite solutions
D.Because we are not given the equations of the lines, it is impossible to know.
Solution & Explanation
Related Formula
Parallel lines have the same slope (m₁ = m₂$m_1 = m_2$) but different y-intercepts (b₁ ≠ b₂$b_1 \neq b_2$).
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 arrow$\rightarrow$ zero intersections arrow$\rightarrow$ no solution.
Chapter Mix
SAT Math: Heart of Algebra
Keywords:#parallel lines#no solution#system of linear equations#coordinate axes
More Heart of Algebra Previous-Year Questions — Page 2
Qsat_math_2023_01_january_morningLinear Equations and Inequalities
Which of the following graphs demonstrates a system of linear equations with no solution?
i.
Four distinct coordinate sub-graphs labeled i, ii, iii, iv showing different line pairs.
ii.
Four distinct coordinate sub-graphs labeled i, ii, iii, iv showing different line pairs.
iii.
Four distinct coordinate sub-graphs labeled i, ii, iii, iv showing different line pairs.
iv.
Four distinct coordinate sub-graphs labeled i, ii, iii, iv showing different line pairs.
Four distinct coordinate sub-graphs labeled i, ii, iii, iv showing different line pairs.
A. i & ii
B. ii & iv
C. iii & iv
D. ii & iii
Solution
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 arrow$\rightarrow$ both represent no solution systems.
Chapter Mix
SAT Math: Heart of Algebra
Qsat_math_2023_01_january_morningSystems of Linear Inequalities
A man buys a$a$ apples and g$g$ oranges, each costing 80 cents and 60 cents respectively. If the man buys less than 18 fruits and spends less than $12, which of the following is the correct inequality representing the above information?
A.a + g = 18$a + g = 18$4a + 3g = 60$4a + 3g = 60$
B.a + g < 18$a + g < 18$4a + 3g < 60$4a + 3g < 60$
C.a + g ≤ 18$a + g \le 18$4a + 3g ≤ 60$4a + 3g \le 60$
D.a + g ≤ 18$a + g \le 18$4a + 3g = 60$4a + 3g = 60$
The system of inequalities is a + g < 18$a + g < 18$ and 4a + 3g < 60$4a + 3g < 60$ (Option B).
Pattern Recognition
'Less than' strictly means <$<$, not ≤$\leq$. Convert dollars to cents before simplifying.
Chapter Mix
SAT Math: Heart of Algebra
Q5sat_math_2023_01_january_morningSystems of Linear Inequalities
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$s$ is the number of sweet M&Ms and b$b$ is the number of bitter M&Ms:
s + b ≥ 25$s + b \geq 25$5s - 10b < 2$5s - 10b < 2$
Which of the following solutions, in the format (s, b)$(s, b)$ could be a possible solution for the number of M&Ms in Samantha's bag?
A.(6, 10)$(6, 10)$
B.(2, 30)$(2, 30)$
C.(1, 1)$(1, 1)$
D.(50, 10)$(50, 10)$
Solution
Related Formula
For a coordinate pair (s, b)$(s, b)$ 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)$(2, 30)$ where s = 2$s = 2$ and b = 30$b = 30$:
Since both statements evaluate to true for (2, 30)$(2, 30)$, option B is a valid solution.
Pattern Recognition
System of inequalities question arrow$\rightarrow$ plug and check given options to find which makes both true.
Chapter Mix
SAT Math: Heart of Algebra
Q9sat_math_2023_01_january_morningSolving Linear Equations
If a + 2b + 3c = 24$a + 2b + 3c = 24$ and 3a + 2b + c = 36$3a + 2b + c = 36$, what is the value of (a + b + c)$(a + b + c)$?
A. 8
B. 12
C. 15
D. 60
Solution
Related Formula
For symmetric linear systems, adding equations directly often isolates the required expression (a + b + c)$(a + b + c)$ without solving for individual variables.
Practice past-year questions one chapter at a time. Pick an exam → subject → chapter and get every PYQ for that topic — pulled together from all past papers — with the chapter's key formulas alongside.
