Solve the following equation for the ratio $\frac{y}{x}$, given that $x \neq 0$: $7(3x - 2y) + 4(x + 5y) = 2(5x + 3y) - 3(x

Solve the following equation for the ratio $\frac{y}{x}$ , given that $x \neq 0$ : $7(3x - 2y) + 4(x + 5y) = 2(5x + 3y) - 3(x - y)$ .

Final answer

$\frac{y}{x} = -6$

Steps

Expand and simplify the left-hand side

To find the ratio $\frac{y}{x}$ , simplify both sides of the equation by expanding the parentheses and combining like terms.

Apply the distributive property to remove the parentheses: $7(3x - 2y) + 4(x + 5y) = 21x - 14y + 4x + 20y$

Group and combine the like terms: $(21x + 4x) + (-14y + 20y) = 25x + 6y$

Expand and simplify the right-hand side

Apply the distributive property to the right-hand side: $2(5x + 3y) - 3(x - y) = 10x + 6y - 3x + 3y$

Group and combine the like terms: $(10x - 3x) + (6y + 3y) = 7x + 9y$

Equate both sides and collect terms

Set the simplified expressions equal to each other: $25x + 6y = 7x + 9y$

Rearrange the equation to isolate the terms involving $x$ and $y$ : $25x - 7x = -(9y - 6y)$ $18x = -3y$

Solve for the ratio $\frac{y}{x}$

Divide both sides of the equation by $-3$ : $-6x = y$

Since $x \neq 0$ , divide both sides by $x$ to find the ratio: $\frac{y}{x} = -6$

Key Takeaway & Common Pitfall: When solving for a ratio of two variables like $\frac{y}{x}$ , it is not necessary to find individual numerical values for $x$ and $y$ . Instead, collect all $x$ -terms on one side and all $y$ -terms on the other side to form a relationship of the form $Ax = By$ , then divide through to obtain $\frac{y}{x} = \frac{A}{B}$ . Take special care with signs when distributing negative coefficients and moving terms across the equals sign.

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student inverts the ratio by computing x/y instead of y/x.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$21x - 14y + 4x + 20y = 10x + 6y - 3x + 3y \implies 25x + 6y = 7x + 9y$
Step 2
$18x = 3y$
Step 3
$\frac{y}{x} = \frac{3}{18} = \frac{1}{6}$
Correct
$\frac{y}{x} = \frac{18}{3} = 6$
From $18x = 3y$ , dividing both sides first by $3$ gives $6x = y$ , and dividing by $x$ gives $\frac{y}{x} = 6$ . You matched the variable in the numerator with its own coefficient, effectively solving for $\frac{x}{y}$ rather than $\frac{y}{x}$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded multiple grouped terms with positive and negative coefficients accurately.

You combined like terms on each side of the equation without making sign mistakes.

You isolated the variable terms into the linear form $18x = 3y$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Ratio Inversion Error

You accidentally placed the coefficient of $y$ in the numerator and the coefficient of $x$ in the denominator, computing $\frac{x}{y}$ instead of $\frac{y}{x}$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the position of coefficients when converting a linear relationship $ax = by$ into a ratio $\frac{y}{x}$ . Dividing both sides by $x$ leaves $y$ divided by $x$ , meaning the coefficient of $x$ ( $a$ ) must be divided by the coefficient of $y$ ( $b$ ), giving $\frac{y}{x} = \frac{a}{b}$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Cross-Multiplication Check: If $\frac{y}{x} = k$ , then cross-multiplying must return your previous line: $y = kx$ .
Multiply your candidate ratio by $x$ to verify that $y = \frac{1}{6}x \implies 6y = x$ , which contradicts $18x = 3y$ .
Two-Step Division Method: To find $\frac{y}{x}$ from $ax = by$ , perform the divisions one step at a time.
First divide by $b$ to isolate $y$ : $y = \frac{a}{b}x$ . Then divide by $x$ : $\frac{y}{x} = \frac{a}{b}$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Divide both sides by $3$ to isolate $y$ : $6x = y$ .
Divide both sides by $x$ (since $x \neq 0$ ): $\frac{y}{x} = 6$ .

Final answer

$\frac{y}{x} = 6$

NEXT STEPS

Final Takeaways from Your Teacher

Practice isolating ratios step-by-step by dividing by the coefficient next to the target numerator first.
Use cross-multiplication as a quick sanity check whenever you write down a final ratio.

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the negative sign to the second term when expanding -3(x - y), writing -3x - 3y instead of -3x + 3y.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
21x - 14y + 4x + 20y = 10x + 6y - 3x - 3y $\implies$ 25x + 6y = 7x + 3y
Correct
21x - 14y + 4x + 20y = 10x + 6y - 3x + 3y $\implies$ 25x + 6y = 7x + 9y
When expanding $-3(x - y)$ , the negative sign must be distributed to both terms inside the parentheses: $(-3) \cdot x + (-3) \cdot (-y) = -3x + 3y$ . Writing $-3x - 3y$ caused the right-hand side to simplify to $7x + 3y$ instead of $7x + 9y$ .
Step 2
18x = -3y
Correct
18x = 3y
You correctly subtracted $7x$ and $6y$ from both sides ( $25x - 7x = 3y - 6y$ ), but this relation carries over the sign error from the first step.
Step 3
$\frac{y}{x} = \frac{18}{-3} = -6$
Correct
$\frac{y}{x} = \frac{18}{3} = 6$
You correctly divided both sides by $x$ and the coefficient of $y$ to express the ratio $\frac{y}{x}$ , but the resulting numerical value is affected by the earlier sign error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded and collected like terms on the left-hand side to obtain $25x + 6y$ .

You correctly rearranged your linear equation to isolate terms in $x$ and $y$ on opposite sides.

You correctly formed and evaluated the ratio $\frac{y}{x}$ from your equation.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Distributive Sign Error

You wrote $-3(x - y)$ as $-3x - 3y$ instead of $-3x + 3y$ , missing the double negative.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Overlooking that multiplying a negative factor by a subtracted term yields a positive term: $(-a) \cdot (-b) = +ab$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Distributing a Negative Factor: For any expression $-a(b - c)$ , the distributed form is $-ab + ac$ .
Circle the negative sign along with the multiplier and explicitly multiply each term inside the parentheses by that signed number before combining terms.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand both sides carefully, ensuring the negative factor distributes to both terms: $21x - 14y + 4x + 20y = 10x + 6y - 3x + 3y$
Combine like terms on each side: $25x + 6y = 7x + 9y$
Collect $x$ -terms on the left and $y$ -terms on the right: $25x - 7x = 9y - 6y \implies 18x = 3y$
Divide both sides by $3x$ (since $x \neq 0$ ) to solve for $\frac{y}{x}$ : $\frac{y}{x} = \frac{18}{3} = 6$

Final answer

$\frac{y}{x} = 6$

NEXT STEPS

Final Takeaways from Your Teacher

Practice distributing negative constants across parentheses containing subtraction, paying special attention to double negatives.
When simplifying multi-term equations, write out intermediate multiplication steps like $(-3)(-y) = +3y$ explicitly before combining.

OverviewACEpath Teacher's Diagnosis

03 The student makes a sign error when moving terms across the equals sign, adding 6y instead of subtracting it.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
$21x - 14y + 4x + 20y = 10x + 6y - 3x + 3y \implies 25x + 6y = 7x + 9y$
Step 2
$18x = 15y$
Correct
$18x = 3y$
When transposing $+6y$ from the left side to the right side, you must apply the inverse operation by subtracting $6y$ . Instead, you added $6y$ to $9y$ to get $15y$ , whereas $9y - 6y = 3y$ .
Step 3
$\frac{y}{x} = \frac{18}{15} = \frac{6}{5}$
Correct
$\frac{y}{x} = \frac{18}{3} = 6$
Your division method to isolate $\frac{y}{x}$ from $18x = 15y$ was algebraically sound, but the numerical answer was affected by the sign error in the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly expanded all expressions using the distributive property, including distributing the negative sign in $-3(x - y)$ .

You accurately combined the variable terms on each side to simplify the equation to $25x + 6y = 7x + 9y$ .

You correctly divided the equation to isolate the target ratio $\frac{y}{x}$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Transposition sign error

You added $6y$ to $9y$ instead of subtracting $6y$ when moving it across the equals sign.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the sign of a term when relocating it across the equals sign, forgetting that moving a positive term requires subtracting it from both sides.

REASONING PATH

Your ACEpath teacher’s suggested approach

Use Inverse Operations: To eliminate an added term $+B$ on one side, subtract $B$ from both sides: $A + B = C \implies A = C - B$ .
Always write out the subtraction step explicitly (e.g., $- 6y$ under both sides) before simplifying.
Ratio Rearrangement Check: For an equation $Ax = By$ , dividing by $Bx$ yields $\frac{y}{x} = \frac{A}{B}$ .
Verify that cross-multiplying $\frac{y}{x} = \frac{A}{B}$ recovers the original balanced line $Ax = By$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Start from the correctly simplified line: $25x + 6y = 7x + 9y$ .
Subtract $7x$ from both sides: $18x + 6y = 9y$ .
Subtract $6y$ from both sides: $18x = 9y - 6y \implies 18x = 3y$ .
Divide both sides by $3x$ : $\frac{y}{x} = \frac{18}{3} = 6$ .

Final answer

$\frac{y}{x} = 6$

NEXT STEPS

Final Takeaways from Your Teacher

Practice two-step linear equations by writing explicit operation arrows beneath terms as you move them across the equals sign.
When solving for a ratio like $\frac{y}{x}$ , perform a quick reverse cross-multiplication check before simplifying your fraction.

Lock it in

Question

Given the equation $ax = by$ with $b \neq 0$ and $x \neq 0$ , what is the correct formula for the ratio $\frac{y}{x}$ ?

Answer

$\frac{y}{x} = \frac{a}{b}$ . The coefficient of $x$ goes to the numerator because cross-multiplying $\frac{y}{x} = \frac{a}{b}$ yields $by = ax$ .

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student inverts the ratio by computing x/y instead of y/x.

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 fails to distribute the negative sign to the second term when expanding -3(x - y), writing -3x - 3y instead of -3x + 3y.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student makes a sign error when moving terms across the equals sign, adding 6y instead of subtracting it.

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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