Q16 (2023)
Using the following equation, what value of $A$ makes the following system of equations have $\left(-\frac{4}{7}, -\frac{1}{7}\right)$ as a solution?
$Ax + 5y = -3$
$-8x + Ay = 4$
- 3
- 5
- 4
- -4
### Related Formula
A point $(x_0, y_0)$ is a solution to a system of equations if substituting $(x_0, y_0)$ into each equation yields a true identity.
### Core Logic
Substitute $x = -\frac{4}{7}$ and $y = -\frac{1}{7}$ into $Ax + 5y = -3$:
$$A\left(-\frac{4}{7}\right) + 5\left(-\frac{1}{7}\right) = -3$$
$$-\frac{4A}{7} - \frac{5}{7} = -3$$
Multiply by 7:
$$-4A - 5 = -21 \implies -4A = -16 \implies A = 4$$
### Step 1: Verification in Second Equation
Check $A = 4$ in $-8x + Ay = 4$:
$$-8\left(-\frac{4}{7}\right) + 4\left(-\frac{1}{7}\right) = \frac{32 - 4}{7} = \frac{28}{7} = 4 \quad \text{(Verified)}$$
Thus, $A = 4$ (Option C).
### Pattern Recognition
Given solution point $(x, y)$ $\rightarrow$ substitute directly into linear equation to solve for unknown parameter $A$.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
SAT Math: Heart of Algebra
