Which of the following graphs demonstrates a system of linear equations with no solution? i.

ii.

iii.

iv.

Related Formula

A point (x₀, y₀)$(x_0, y_0)$ is a solution to a system of equations if substituting (x₀, y₀)$(x_0, y_0)$ into each equation yields a true identity.

Core Logic

Substitute x = -(4)/(7)$x = -\frac{4}{7}$ and y = -(1)/(7)$y = -\frac{1}{7}$ into Ax + 5y = -3$Ax + 5y = -3$:

Multiply by 7:

Step 1: Verification in Second Equation

Check A = 4$A = 4$ in -8x + Ay = 4$-8x + Ay = 4$:

Thus, A = 4$A = 4$ (Option C).

Pattern Recognition

Given solution point (x, y)$(x, y)$ arrow$\rightarrow$ substitute directly into linear equation to solve for unknown parameter A$A$.

Chapter Mix

SAT Math: Heart of Algebra

Related Formula

For y < mx + b$y < mx + b$, boundary is dashed, shading is below. For y ≥ mx + b$y \geq mx + b$, boundary is solid, shading is above.

Core Logic

Analyze boundary line 1: y = 2x + 4$y = 2x + 4$

• Dashed line with y-intercept (0, 4)$(0, 4)$ and positive slope 2$2$.
• y < 2x + 4$y < 2x + 4 \implies$ shade BELOW the dashed line.

Analyze boundary line 2: y = -(1)/(3)x - 2$y = -\frac{1}{3}x - 2$

• Solid line with y-intercept (0, -2)$(0, -2)$ and negative slope -(1)/(3)$-\frac{1}{3}$.
• y ≥ -(1)/(3)x - 2$y \geq -\frac{1}{3}x - 2 \implies$ shade ABOVE the solid line.

Step 1: Select Matching Graph

Graph A correctly shows shading above the solid line y = -(1)/(3)x - 2$y = -\frac{1}{3}x - 2$ and below the dashed line y = 2x + 4$y = 2x + 4$.

Hence, choice A is correct.

Pattern Recognition

Dashed vs. solid line test arrow$\rightarrow$ strict inequality (<$<$) requires dashed; non-strict (≥$\geq$) requires solid.

Chapter Mix

SAT Math: Heart of Algebra