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 & Explanation
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
Keywords:#multiple system graphs#parallel lines#no solution#coordinate grid
More Heart of Algebra Previous-Year Questions
Qsat_math_2023_01_january_morningSlope of Perpendicular Lines
What is the slope of the line perpendicular to the one shown below?
A Cartesian coordinate plane showing a straight line passing through (0, -3) and (4, 0).
A.(3)/(4)$\frac{3}{4}$
B.(4)/(3)$\frac{4}{3}$
C.-(4)/(3)$-\frac{4}{3}$
D.-(3)/(4)$-\frac{3}{4}$
Solution
Related Formula
m⊥ = -(1)/(m)$$m_{\perp} = -\frac{1}{m}$$
where m$m$ is the slope of the given line and m⊥$m_{\perp}$ is the slope of the line perpendicular to it.
Core Logic
Identify two points on the line shown in the graph: (0, -3)$(0, -3)$ and (4, 0)$(4, 0)$.
Calculate the slope m$m$ using the rise over run formula:
Line graph arrow$\rightarrow$ count rise/run to find m$m$. Perpendicular slope is always the negative reciprocal flip.
Chapter Mix
SAT Math: Heart of Algebra
Qsat_math_2023_01_january_morningLinear Function Word Problems
Mark runs a tutoring service where he charges $\text{\$7}for each new client and$ for each new client and $\text{\20$20}$ per hour. His new client gets 4 hours of tutoring and pays Mark $\text{\$100}$. Which of the following statements is correct?
A. Mark was overpaid
B. Mark was underpaid
C. You cannot determine if he was overpaid or underpaid based on the information given.
D. Mark was paid the correct amount for his services.
Since $\100 >$100 > \$87, Mark received$, Mark received $\13$13$ more than his standard charge. Therefore, Mark was overpaid.
Pattern Recognition
Word problem with fixed + variable cost arrow$\rightarrow$ set up total equation, evaluate, compare to actual payout.
Chapter Mix
SAT Math: Heart of Algebra
Qsat_math_2023_01_january_morningGraphing Linear Equations
Which of the following lines has an undefined slope?
A coordinate graph displaying four lines labeled Red Line, Green Line, Blue Line, and Orange Line.
A. Red line
B. Green line
C. Blue line
D. Orange line
Solution
Related Formula
m = (Δ y)/(Δ x)$$m = \frac{\Delta y}{\Delta x}$$
Core Logic
A vertical line has a constant x$x$-value, which means Δ x = 0$\Delta x = 0$.
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$x = 2$.
Step 1: Conclusion
Therefore, the Blue line has an undefined slope.
Pattern Recognition
Horizontal line arrow$\rightarrow$ slope = 0. Vertical line arrow$\rightarrow$ slope is undefined.
Chapter Mix
SAT Math: Heart of Algebra
Qsat_math_2023_01_january_morningLinear Function Word Problems
Megan and her friends all go to buy candy at the store. Mark pays \15 for 3 chocolate bars, Megan 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$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 $yrepresents total price and$ represents total price and $x$ represents number of chocolate bars.
A.5y = x$5y = x$
B.y = 5x$y = 5x$
C.y = x + 5$y = x + 5$
D.15y = 3x$15y = 3x$
Solution
Related Formula
y = kx$y = kx$
where k$k$ represents the unit price per item, x$x$ represents quantity, and y$y$ represents total cost.
Core Logic
Determine the unit cost per chocolate bar from each order:
Proportional direct variation arrow$\rightarrow$ total = unit rate ×$\times$ quantity.
Chapter Mix
SAT Math: Heart of Algebra
Qsat_math_2023_01_january_morningSolving Systems of Linear Equations
What is the solution to the system of linear equations demonstrated in the graph?
A graph showing two intersecting lines meeting at (2, 0) on the x-axis.
A.(0, 2)$(0, 2)$
B.(0, 0)$(0, 0)$
C.(2, 0)$(2, 0)$
D.(2, 2)$(2, 2)$
Solution
Related Formula
The solution to a system of linear equations represented graphically is given by the point of intersection (x, y)$(x, y)$ 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$x = 2$ and y = 0$y = 0$.
Step 1: Conclusion
Thus, the point of intersection is (2, 0)$(2, 0)$, corresponding to option C.
Pattern Recognition
Graph of system arrow$\rightarrow$ solution is simply the intersection point.
Chapter Mix
SAT Math: Heart of Algebra
More Heart of Algebra Questions — sat_math_2023_01_january_morning
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.
