Worked Examples: SAT Strategy
Example 1: Easy (Standard Form Intercepts)
Question: The graph of the linear equation $3x - 5y = 30$ in the $xy$-plane has an $x$-intercept at $(a, 0)$ and a $y$-intercept at $(0, b)$. What is the value of $a + b$?
SAT Strategy: Do NOT rearrange the equation into $y = mx + c$. In Standard Form ($Ax + By = C$), you can find the intercepts instantly by "covering up" the other variable (setting it to 0).
- To find the $x$-intercept ($a$), cover up the $y$-term: $3x = 30 \implies x = 10$. So, $a = 10$.
- To find the $y$-intercept ($b$), cover up the $x$-term: $-5y = 30 \implies y = -6$. So, $b = -6$.
- Add them together: $10 + (-6) = 4$.
- Answer: $4$.
Example 2: Medium (Perpendicular Lines & Abstract Variables)
Question: In the $xy$-plane, Line $k$ is defined by the equation $cx + 4y = 12$, where $c$ is a constant. Line $j$ is perpendicular to Line $k$ and passes through the origin. If Line $j$ contains the point $(3, -5)$, what is the value of $c$?
SAT Strategy: Use the slope formula $m = \frac{\Delta y}{\Delta x}$ for Line $j$, then use the negative reciprocal rule ($m_1 \times m_2 = -1$) to find the slope of Line $k$.
- Line $j$ passes through $(0,0)$ and $(3, -5)$. Its slope is $\frac{-5 - 0}{3 - 0} = -\frac{5}{3}$.
- Because Line $k$ is perpendicular, its slope must be the negative reciprocal: $\frac{3}{5}$.
- Convert $cx + 4y = 12$ into slope-intercept form to reveal its slope: $4y = -cx + 12 \implies y = -\frac{c}{4}x + 3$.
- Set the slope equal to $\frac{3}{5}$: $-\frac{c}{4} = \frac{3}{5} \implies -5c = 12 \implies c = -2.4$.
- Answer: $-2.4$.
Example 3: Hard (Shifting Lines / Translations)
Question: The function $f(x)$ is defined by $f(x) = 2x + 7$. The graph of $y = g(x)$ in the $xy$-plane is the result of translating the graph of $f$ down $4$ units and right $3$ units. What is the $x$-intercept of the graph of $g$?
SAT Strategy: As an AA SL student, you know function transformations. A shift down by $4$ and right by $3$ means $g(x) = f(x - 3) - 4$. Apply this directly to the linear equation.
- Substitute $(x - 3)$ into the function and subtract $4$: $g(x) = 2(x - 3) + 7 - 4$.
- Simplify: $g(x) = 2x - 6 + 3 \implies g(x) = 2x - 3$.
- The question asks for the $x$-intercept. Set $g(x) = 0$: $0 = 2x - 3 \implies 2x = 3 \implies x = 1.5$.
- Answer: $1.5$.
Practice Questions
Level 1: Easy (Foundational SAT)
- The equation of a line in the $xy$-plane is $y = -3x + 12$. What is the $x$-coordinate of the $x$-intercept of the line? (Grid-in)
- A line in the $xy$-plane passes through the points $(0, 4)$ and $(2, 10)$. Which of the following is an equation of the line?
- A) $y = 3x + 4$
- B) $y = 4x + 2$
- C) $y = 3x + 2$
- D) $y = 2x + 4$
- The graph of $4x + 5y = 40$ in the $xy$-plane has a $y$-intercept at $(0, b)$. What is the value of $b$? (Grid-in)
- Line $p$ is defined by the equation $y = \frac{2}{3}x - 5$. Line $q$ is parallel to line $p$ and passes through the origin. What is the equation of line $q$?
- A) $y = -\frac{3}{2}x$
- B) $y = \frac{2}{3}x$
- C) $y = \frac{2}{3}x + 5$
- D) $y = -\frac{3}{2}x - 5$
- A line has a slope of $-2$ and passes through the point $(4, 1)$. What is the $y$-intercept of this line? (Grid-in)
Level 2: Medium (Standard SAT)
- Line $l$ is defined by the equation $2x - 7y = 14$. What is the slope of a line that is perpendicular to line $l$?
- A) $-\frac{7}{2}$
- B) $-\frac{2}{7}$
- C) $\frac{2}{7}$
- D) $\frac{7}{2}$
- The function $f$ is defined by $f(x) = -\frac{1}{4}x + 6$. The graph of $y = f(x)$ is shifted up 3 units to create the graph of $y = g(x)$. What is the $x$-intercept of the graph of $g$? (Grid-in)
- A line passes through the points $(-3, a)$ and $(5, a+16)$. What is the slope of the line?
- A) $0.5$
- B) $2$
- C) $8$
- D) It cannot be determined from the given information.
- The line $y = mx + b$ passes through the points $(-2, -5)$ and $(3, 10)$. What is the value of $m + b$? (Grid-in)
- The graph of $kx - 3y = 18$ has an $x$-intercept of $6$. What is the value of the constant $k$? (Grid-in)
Level 3: Hard (Advanced SAT)
- Line $j$ in the $xy$-plane passes through the points $(a, 0)$ and $(0, b)$, where $a$ and $b$ are positive constants. What is the slope of line $j$ in terms of $a$ and $b$?
- A) $\frac{a}{b}$
- B) $\frac{b}{a}$
- C) $-\frac{a}{b}$
- D) $-\frac{b}{a}$
- The line $3x - 4y = 24$ forms a right triangle with the $x$-axis and the $y$-axis in the $xy$-plane. What is the area of this triangle? (Grid-in)
- The function $f$ is defined by $f(x) = 5x - 2$. The graph of $g(x)$ is created by applying the transformation $g(x) = f(x - 3) + 4$. What is the $y$-intercept of $g(x)$? (Grid-in)
- In the $xy$-plane, line $L_1$ is given by $ax + by = c$, where $a, b,$ and $c$ are strictly positive constants. Which quadrant does line $L_1$ NOT pass through?
- A) Quadrant I
- B) Quadrant II
- C) Quadrant III
- D) Quadrant IV
- Line $A$ has the equation $y = 3x - 8$. Line $B$ is perpendicular to Line $A$ and intersects Line $A$ at the point $(4, 4)$. What is the $y$-intercept of Line $B$? (Grid-in)
Level 4: Very Hard (Slightly beyond SAT)
- The line $8x + 15y = 120$ intersects the $x$-axis at point $P$ and the $y$-axis at point $Q$. What is the straight-line distance between point $P$ and point $Q$? (Grid-in)
- Line $M$ has an $x$-intercept of $4$ and a $y$-intercept of $6$. Line $N$ has an $x$-intercept of $6$ and a $y$-intercept of $4$. The two lines intersect at the point $(x, y)$. What is the value of $x + y$? (Grid-in)
- Two perpendicular lines intersect at the point $(2, 5)$. If the sum of their slopes is $\frac{8}{3}$ and one of the lines has a positive slope, what is the slope of the steeper line?
- A) $\frac{1}{3}$
- B) $3$
- C) $4$
- D) $5$
- The line defined by $y = mx + b$ passes through the point $(p, q)$, where $p$ and $q$ are non-zero constants. If the line is shifted to the left by $p$ units and down by $q$ units, the new line passes through the origin. Which of the following must be true?
- A) $b = 0$
- B) $m = 1$
- C) $q = mp + b$
- D) The original line already passed through the origin.
- The graph of $y = a|x - 2| + c$ intersects the $x$-axis at exactly two points, $(0, 0)$ and $(4, 0)$. It intersects the $y$-axis at exactly one point. If $a$ and $c$ are constants, what is the value of $\frac{c}{a}$? (Grid-in)
🔒 Answer Key & Brief Explanations
Sign in with your subscribed email to view worked reasoning for every question. Understanding why each step works is the point — not just the letter answer.
Unlock full solutions →