In coordinate geometry and vector statics, two intersecting lines are described by the symmetric linear system parameterization $\frac{x - x_0}{a} - \frac{y - y_0}{b} = c$, where $a, b \neq 0$. Solve this literal equation for $y$, taking care to preserve the correct distribution of negative signs across all terms in the numerator.

In coordinate geometry and vector statics, two intersecting lines are described by the symmetric linear system parameterization $\frac{x - x_0}{a} - \frac{y - y_0}{b} = c$ , where $a, b \neq 0$ . Solve this literal equation for $y$ , taking care to preserve the correct distribution of negative signs across all terms in the numerator.

Final answer

$y = y_0 + \frac{b(x - x_0) - abc}{a}$

Steps

To solve the literal equation for $y$ , isolate the term containing $y$ step by step using standard algebraic operations.

Step 1: Isolate the fraction containing $y$

Start with the given equation: $\frac{x - x_0}{a} - \frac{y - y_0}{b} = c$

Add $\frac{y - y_0}{b}$ to both sides and subtract $c$ from both sides: $\frac{y - y_0}{b} = \frac{x - x_0}{a} - c$

Common Pitfall: If you instead subtract $\frac{x - x_0}{a}$ from both sides, remember that the leading minus sign belongs to the entire fraction: $-\frac{y - y_0}{b}$ . Multiplying across by $-1$ or moving the fraction to the other side early avoids sign errors.

Step 2: Combine the terms on the right-hand side over a common denominator

Write $c$ with the denominator $a$ : $c = \frac{ac}{a}$

Substitute this back into the right side: $\frac{y - y_0}{b} = \frac{(x - x_0) - ac}{a}$

Step 3: Clear the denominator $b$

Multiply both sides of the equation by $b$ : $y - y_0 = b \cdot \left(\frac{x - x_0 - ac}{a}\right)$

Distribute $b$ across the numerator: $y - y_0 = \frac{b(x - x_0) - abc}{a}$

Step 4: Solve for $y$

Add $y_0$ to both sides: $y = y_0 + \frac{b(x - x_0) - abc}{a}$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets that the negative sign in front of the fraction applies to the entire numerator -(y - y₀), treating it as -y - y₀ instead of -y + y₀.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$-\frac{(y - y_0)}{b} = c - \frac{x - x_0}{a}$
Step 2
$\frac{(-y - y_0)}{b} = c - \frac{x - x_0}{a}$
Correct
$\frac{-y + y_0}{b} = c - \frac{x - x_0}{a}$
When moving the negative sign into the numerator, it must distribute across every term: $-(y - y_0) = -y - (-y_0) = -y + y_0$ . The sign of $y_0$ was not flipped.
Step 3
$-y - y_0 = bc - \frac{b(x - x_0)}{a}$
Correct
$-y + y_0 = bc - \frac{b(x - x_0)}{a}$
You correctly multiplied both sides of the equation by $b$ , but this step carries forward the sign error from the previous line.
Step 4
$-y = y_0 + bc - \frac{b(x - x_0)}{a} \implies y = -y_0 - bc + \frac{b(x - x_0)}{a}$
Correct
$-y = -y_0 + bc - \frac{b(x - x_0)}{a} \implies y = y_0 - bc + \frac{b(x - x_0)}{a}$
You correctly performed the operations to isolate $y$ (adding $y_0$ then negating both sides), but the final value has incorrect signs due to the earlier distribution error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly isolated the fraction containing $y$ on one side of the equation.

You maintained the correct grouped structure $(x - x_0)$ when multiplying both sides by $b$ .

You correctly solved for $y$ from $-y$ by negating all terms across both sides.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error across fraction numerator

Distributing the negative sign into the numerator as $-y - y_0$ rather than $-y + y_0$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating the leading negative sign of a fraction as attaching solely to the first term $y$ rather than distributing across the entire grouped binomial $(y - y_0)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Fraction Bar as Grouping Symbol: $-\frac{A - B}{C} = \frac{-(A - B)}{C} = \frac{-A + B}{C}$
Always place parentheses around a multi-term numerator before applying a leading minus sign.
Alternative: Add the Entire Fraction First: If a term has a negative coefficient or minus sign, adding it to the opposite side eliminates negative sign distribution entirely.
Rewrite $\frac{x - x_0}{a} - \frac{y - y_0}{b} = c$ as $\frac{x - x_0}{a} - c = \frac{y - y_0}{b}$ to keep the numerator positive.

CORRECTED PATH

ACEpath Teacher's Demonstration

Rewrite the left-hand numerator: $-(y - y_0) = -y + y_0$ .
Multiply both sides by $b$ : $-y + y_0 = bc - \frac{b(x - x_0)}{a}$ .
Subtract $y_0$ from both sides: $-y = -y_0 + bc - \frac{b(x - x_0)}{a}$ .
Multiply both sides by $-1$ : $y = y_0 - bc + \frac{b(x - x_0)}{a}$ .

Final answer

$y = y_0 + \frac{b(x - x_0) - abc}{a}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice inserting explicit parentheses around any multi-term numerator when a minus sign precedes the fraction.
Try isolating negative terms by moving them to the other side of the equals sign to minimize minus-sign errors.
Work through the flashcards and practice problems below to master distributing negative signs over binomial numerators.

OverviewACEpath Teacher's Diagnosis

02 The student multiplies both sides by b but forgets the leading negative sign on the fraction term, dropping the negative sign entirely.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{x - x_0}{a}$ - $\frac{y - y_0}{b}$ = c $\implies$ $\frac{y - y_0}{b}$ = c - $\frac{x - x_0}{a}$
Correct
- $\frac{y - y_0}{b}$ = c - $\frac{x - x_0}{a}$ $\implies$ $\frac{y - y_0}{b}$ = $\frac{x - x_0}{a}$ - c
When you subtracted $\frac{x - x_0}{a}$ from both sides, the leading minus sign attached to the fraction was omitted. The left-hand side is $-\frac{y - y_0}{b}$ , which reverses the signs on the right side when multiplied by $-1$ .
Step 2
y - y_0 = bc - $\frac{b(x - x_0)}{a}$
Correct
y - y_0 = $\frac{b(x - x_0)}{a}$ - bc
You correctly multiplied every term on both sides by $b$ , but the expression inherits the inverted signs from the previous step.
Step 3
y = y_0 + bc - $\frac{b(x - x_0)}{a}$
Correct
y = y_0 + $\frac{b(x - x_0) - abc}{a}$
You correctly added $y_0$ to isolate $y$ , but the final answer retains the reversed signs from Step 1.

WHAT YOU DID WELL

Compliments from Your Teacher

You clearly laid out each algebraic operation line by line.

You correctly cleared the denominator $b$ by distributing multiplication across all terms.

You added $y_0$ to both sides correctly in your final step to isolate $y$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Dropped leading negative sign on fraction term

When subtracting $\frac{x - x_0}{a}$ from both sides, the minus sign in front of $\frac{y - y_0}{b}$ was lost.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating subtraction as a detachable separator rather than an operational sign bound to the following fraction term ( $-\frac{A}{B}$ ).

REASONING PATH

Your ACEpath teacher’s suggested approach

Keep the sign with the term: In $A - B = C$ , subtracting $A$ yields $-B = C - A$ , not $B = C - A$ .
Circle the sign immediately preceding the term before moving other terms across the equals sign.
Alternative: add the negative term: Adding a subtracted term to the opposite side keeps the coefficient positive: $A - B = C \implies A = B + C$ .
Whenever the variable you want to isolate has a leading minus sign, add that entire term to the opposite side first.

CORRECTED PATH

ACEpath Teacher's Demonstration

Start from $\frac{x - x_0}{a} - \frac{y - y_0}{b} = c$ .
Add $\frac{y - y_0}{b}$ to both sides and subtract $c$ : $\frac{y - y_0}{b} = \frac{x - x_0}{a} - c$ .
Express the right-hand side over a common denominator: $\frac{y - y_0}{b} = \frac{(x - x_0) - ac}{a}$ .
Multiply both sides by $b$ : $y - y_0 = \frac{b(x - x_0) - abc}{a}$ .
Add $y_0$ to both sides to solve for $y$ .

Final answer

y = y_0 + $\frac{b(x - x_0) - abc}{a}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice moving subtracted terms to the other side first so you never have to track a leading negative coefficient.
Whenever you subtract a term from an equation with a minus sign in front of the target, explicitly write $-1 \cdot (\dots)$ before proceeding.
Check your sign distribution by substituting simple test values like $x = x_0$ into both the original equation and your final formula.

OverviewACEpath Teacher's Diagnosis

03 When subtracting y₀ after isolating -y, the student subtracts instead of adding, making a sign error in the final step.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$\frac{y - y_0}{b}$ = $\frac{x - x_0}{a}$ - c
Step 2
y - y_0 = $\frac{b(x - x_0)}{a}$ - bc
Step 3
y = -y_0 + $\frac{b(x - x_0)}{a}$ - bc
Correct
y = y_0 + $\frac{b(x - x_0)}{a}$ - bc
To undo subtracting $y_0$ in $y - y_0$ , you need to perform the inverse operation by adding $y_0$ to both sides, which yields $+y_0$ , not $-y_0$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly isolated the fraction containing the target variable.

You distributed the factor $b$ across all terms on the right-hand side without error.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Inverse Operation Sign Error

You moved the subtracted term $-y_0$ to the right side without reversing its sign to $+y_0$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the addition property of equality when isolating a variable, keeping the negative sign of a subtracted constant when transposing it to the opposite side of the equation.

REASONING PATH

Your ACEpath teacher’s suggested approach

Use Inverse Operations to Isolate: If an expression has $y - k$ , undo the subtraction by adding $k$ to both sides: $y - k + k = y$ .
Always write out $+ y_0$ on both sides explicitly before simplifying the final expression.

CORRECTED PATH

ACEpath Teacher's Demonstration

Add $y_0$ to both sides of the equation: $y = y_0 + \frac{b(x - x_0)}{a} - bc$ .
Optionally express over a single common denominator $a$ : $y = y_0 + \frac{b(x - x_0) - abc}{a}$ .

Final answer

y = y_0 + $\frac{b(x - x_0)}{a}$ - bc $\text{or}$ y = y_0 + $\frac{b(x - x_0) - abc}{a}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice checking inverse operations by substituting a simple number for each variable to verify that both sides remain balanced.
Pay special attention to signs when moving terms across the equals sign.

Lock it in

Question

How do you distribute the negative sign into the numerator of $-\frac{u - v}{w}$ ?

Answer

$\frac{-u + v}{w}$ . The fraction bar acts as parentheses around the numerator: $-(u - v) = -u + v$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets that the negative sign in front of the fraction applies to the entire numerator -(y - y₀), treating it as -y - y₀ instead of -y + y₀.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student multiplies both sides by b but forgets the leading negative sign on the fraction term, dropping the negative sign entirely.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 When subtracting y₀ after isolating -y, the student subtracts instead of adding, making a sign error in the final step.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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