Converting Between Standard and Slope-Intercept Form
Standard form $Ax + By = C$ and slope-intercept form $y = mx + b$ describe the same line. Converting is just careful algebra.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
To get slope-intercept form, solve for $y$: move the $x$-term across, then divide every term by the coefficient of $y$. Watch the sign when the $x$-term changes sides.
How to solve one, step by step
Example: rewrite $2x + 3y = 12$.
- Subtract $2x$: $3y = -2x + 12$.
- Divide all terms by 3: $y = -\frac{2}{3}x + 4$.
The mistakes that cost points
- Dividing only some terms. Every term must be divided by the coefficient of $y$, not just the $x$-term.
- Sign error moving the x-term. Moving $2x$ to the other side makes it $-2x$.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A recipe requires $3$ cups of oats for every $k$ dozen cookies produced. A baker makes $4$ dozen cookies using $12$ cups of oats. What is the value of $k$?
- A$3$
- B$1$
- C$12$
- D$4$
Show solution
Answer: B, $1$. The ratio of cups to dozens is constant: $\frac{3}{k} = \frac{12}{4}$. Cross-multiply: $12k = 12 \Rightarrow k = 1$. So $3$ cups are needed per dozen cookies.
Medium
A tank is being filled with water at a rate of $4$ gallons per minute and drained at a rate of $k$ gallons per minute simultaneously. After $10$ minutes, the net gain is $20$ gallons. What is the value of $k$?
- A$\frac{1}{5}$
- B$2$
- C$20$
- D$6$
Show solution
Answer: B, $2$. Net gain per minute $= 4 - k$. Total net gain in $10$ minutes $= 10(4 - k) = 20$. Divide by $10$: $4 - k = 2 \Rightarrow k = 2$ gallons per minute.
Hard
A chemist has a $k$-liter solution that is $40\%$ salt and a $3$-liter solution that is $10\%$ salt. The chemist combines the two solutions. The resulting mixture is $25\%$ salt. What is the value of $k$?
- A$3$
- B$6$
- C$\frac{3}{2}$
- D$\frac{9}{4}$
Show solution
Answer: A, $3$. Salt from first solution: $0.40k$. Salt from second solution: $0.10 \times 3 = 0.30$. Total volume: $k + 3$. Set up the concentration equation: $\frac{0.40k + 0.30}{k + 3} = 0.25$. Multiply both sides by $(k + 3)$: $0.40k + 0.30 = 0.25k + 0.75 \Rightarrow 0.15k = 0.45 \Rightarrow k = 3$ liters.
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