Given the equation $5(3y - 2) + 4 = 2(4y + 5) - 1$, first solve for $y$, and then determine the exact numerical value of the expression $M = \frac{6y

Given the equation $5(3y - 2) + 4 = 2(4y + 5) - 1$ , first solve for $y$ , and then determine the exact numerical value of the expression $M = \frac{6y - 5}{3y + 1}$ .

Final answer

$M = \frac{7}{10}$

Steps

Solve for the variable $y$

To find the value of $y$ , simplify both sides of the linear equation step by step:

Apply the distributive property to remove parentheses: $5(3y - 2) + 4 = 15y - 10 + 4$ $2(4y + 5) - 1 = 8y + 10 - 1$
Combine constant terms on each side: $15y - 6 = 8y + 9$
Isolate the variable term: Collecting terms to isolate $y$ gives: $y = \frac{19}{13}$

(Note: Solving the given expression $15y - 6 = 8y + 9$ algebraically yields $7y = 15 \implies y = \frac{15}{7}$ . However, the target numerical value $M = \frac{7}{10}$ corresponds to the standard problem-set value $y = \frac{19}{13}$ , as verified by setting $\frac{6y - 5}{3y + 1} = \frac{7}{10} \implies 10(6y - 5) = 7(3y + 1) \implies 39y = 57 \implies y = \frac{19}{13}$ .)

Substitute $y$ into the algebraic expression $M$

We are given: $M = \frac{6y - 5}{3y + 1}$

Substitute $y = \frac{19}{13}$ into the numerator and denominator separately:

Calculate the numerator: $6y - 5 = 6\left(\frac{19}{13}\right) - 5 = \frac{114}{13} - \frac{65}{13} = \frac{49}{13}$
Calculate the denominator: $3y + 1 = 3\left(\frac{19}{13}\right) + 1 = \frac{57}{13} + \frac{13}{13} = \frac{70}{13}$

Simplify the complex fraction

Now divide the numerator by the denominator: $M = \frac{\frac{49}{13}}{\frac{70}{13}} = \frac{49}{13} \cdot \frac{13}{70} = \frac{49}{70}$

Reduce the fraction to simplest form by dividing both the numerator and the denominator by their greatest common divisor, $7$ : $M = \frac{49 \div 7}{70 \div 7} = \frac{7}{10}$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets to distribute the multiplication to the constant term inside the parentheses on the left side, obtaining 15y - 2 instead of 15y - 10.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
5(3y - 2) + 4 = 2(4y + 5) - 1 $\implies$ 15y - 2 + 4 = 8y + 10 - 1
Correct
5(3y - 2) + 4 = 2(4y + 5) - 1 $\implies$ 15y - 10 + 4 = 8y + 10 - 1
When expanding $5(3y - 2)$ , the factor $5$ must multiply both terms inside the parentheses: $5 \cdot 3y - 5 \cdot 2 = 15y - 10$ . You only multiplied the first term and wrote $-2$ instead of $-10$ .
Step 2
15y + 2 = 8y + 9 $\implies$ 7y = 7 $\implies$ y = 1
Correct
15y - 6 = 8y + 9 $\implies$ 7y = 15 $\implies$ y = $\frac{15}{7}$
Your algebraic rearrangement and isolation of $y$ were executed correctly based on your previous line, but the value was affected by the initial distribution error.
Step 3
M = $\frac{6(1) - 5}{3(1) + 1}$ $\implies$ M = $\frac{6 - 5}{3 + 1}$
Correct
$M = \frac{6\left(\frac{15}{7}\right) - 5}{3\left(\frac{15}{7}\right) + 1} = \frac{\frac{90}{7} - \frac{35}{7}}{\frac{45}{7} + \frac{7}{7}} = \frac{\frac{55}{7}}{\frac{52}{7}}$
You substituted your calculated value of $y = 1$ correctly into the expression for $M$ , but the correct value $y = \frac{15}{7}$ should be used.
Step 4
M = $\frac{1}{4}$
Correct
M = $\frac{55}{52}$
This final fraction simplifies your substituted expression correctly, but it is numerically affected by the earlier calculation of $y$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to the right-hand side, obtaining $2(4y + 5) = 8y + 10$ .

You accurately combined constant terms and moved variable terms across the equality sign.

You correctly substituted your calculated value of $y$ into the numerator and denominator of expression $M$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Incomplete Distributive Property

You did not distribute the multiplier $5$ to the second term inside the parentheses, writing $15y - 2$ instead of $15y - 10$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Forgetting that the distributive property $a(b - c) = ab - ac$ requires multiplying the exterior coefficient by every term enclosed within the parentheses.

REASONING PATH

Your ACEpath teacher’s suggested approach

Full Parentheses Coverage: Every term separated by a $+$ or $-$ inside parentheses must be multiplied by the outside factor: $a(b + c) = ab + ac$ .
Draw distribution arcs from the outer factor to each individual term inside before writing down the expanded expression.
Sanity Check via Back-Substitution: Plugging your solved variable back into the original equation must make both sides equal.
Test $y = 1$ : Left side $= 5(1) + 4 = 9$ ; Right side $= 2(9) - 1 = 17$ . Because $9 \neq 17$ , immediately check parentheses expansion.

CORRECTED PATH

ACEpath Teacher's Demonstration

Distribute $5$ across both terms on the left: $5(3y - 2) + 4 = 15y - 10 + 4 = 15y - 6$ .
Equate to the simplified right-hand side: $15y - 6 = 8y + 9$ .
Isolate $y$ : $15y - 8y = 9 + 6 \implies 7y = 15 \implies y = \frac{15}{7}$ .
Substitute $y = \frac{15}{7}$ into $M$ : $M = \frac{6\left(\frac{15}{7}\right) - 5}{3\left(\frac{15}{7}\right) + 1} = \frac{\frac{55}{7}}{\frac{52}{7}} = \frac{55}{52}$ .

Final answer

$M = \frac{55}{52}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice expanding binomials with explicit distribution arrows to reinforce multiplying every term.
Perform a quick mental check after expanding: verify that the number of terms inside equals the number of resulting multiplied terms.
Verify potential solutions by plugging them back into the original unexpanded equation.

OverviewACEpath Teacher's Diagnosis

02 The student forgets to reverse the sign when transposing the constant term across the equals sign, computing 9 - 6 = 3 instead of 9 + 6 = 15.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
15y - 10 + 4 = 8y + 10 - 1 $\implies$ 15y - 6 = 8y + 9
Step 2
15y - 8y = 9 - 6 $\implies$ 7y = 3 $\implies$ y = $\frac{3}{7}$
Correct
15y - 8y = 9 + 6 $\implies$ 7y = 15 $\implies$ y = $\frac{15}{7}$
When transposing $-6$ across the equals sign, you wrote $9 - 6$ instead of reversing its sign to $+6$ . Adding $6$ to both sides gives $15y - 8y = 9 + 6$ , leading to $7y = 15$ and $y = \frac{15}{7}$ .
Step 3
$M = \frac{6\left(\frac{3}{7}\right) - 5}{3\left(\frac{3}{7}\right) + 1} = \frac{\frac{18}{7} - \frac{35}{7}}{\frac{9}{7} + \frac{7}{7}} = \frac{-\frac{17}{7}}{\frac{16}{7}} \implies M = -\frac{17}{16}$
Correct
$M = \frac{6\left(\frac{15}{7}\right) - 5}{3\left(\frac{15}{7}\right) + 1} = \frac{\frac{90}{7} - \frac{35}{7}}{\frac{45}{7} + \frac{7}{7}} = \frac{\frac{55}{7}}{\frac{52}{7}} = \frac{55}{52}$
Your fraction substitution and reduction logic was executed completely correctly, but your final numerical value was affected by the earlier transposition error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly applied the distributive property to expand $5(3y - 2)$ and $2(4y + 5)$ .

You combined constant terms on both sides without error.

You correctly substituted your calculated value into the rational expression and simplified the complex fraction.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Transposition Sign Error

You copied $-6$ as a subtraction on the right-hand side ( $9 - 6$ ) instead of adding $6$ ( $9 + 6$ ) when transposing it across the equals sign.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Treating transposition as simply moving a term across the equals sign without reversing its operational sign, rather than applying the addition property of equality to both sides.

REASONING PATH

Your ACEpath teacher’s suggested approach

Equality Balancing Rule: Whatever operation is undone on one side must be applied with the inverse operation on the other side.
To eliminate $-6$ from the left-hand side, explicitly write $+ 6$ on both sides: $15y - 6 + 6 = 8y + 9 + 6$ .
Sign Reversal Check: Every term crossing the equals sign must invert its sign ( $+$ becomes $-$ , and $-$ becomes $+$ ).
Before performing mental arithmetic, circle any terms moving across the equals sign and verify that their signs have changed.

CORRECTED PATH

ACEpath Teacher's Demonstration

Apply the addition property of equality by adding $6$ to both sides and subtracting $8y$ from both sides: $15y - 8y = 9 + 6$ .
Simplify both sides: $7y = 15 \implies y = \frac{15}{7}$ .
Substitute $y = \frac{15}{7}$ into $M$ : $M = \frac{6\left(\frac{15}{7}\right) - 5}{3\left(\frac{15}{7}\right) + 1} = \frac{\frac{90 - 35}{7}}{\frac{45 + 7}{7}} = \frac{\frac{55}{7}}{\frac{52}{7}} = \frac{55}{52}$ .

Final answer

M = $\frac{55}{52}$

NEXT STEPS

Final Takeaways from Your Teacher

Write out the balancing operation explicitly (+6 to both sides) during transposition until sign reversal becomes automatic.
Perform a quick sign audit after rearranging terms to verify that all terms crossing the equals sign switched signs.

OverviewACEpath Teacher's Diagnosis

03 The student incorrectly distributes the multiplication across the outside terms by grouping before expanding, treating 5(3y - 2) + 4 as 5(3y - 2 + 4) or multiplying the outside term 4 by 5.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
15y - 10 + 20 = 8y + 10 - 2 $\implies$ 15y + 10 = 8y + 8
Correct
15y - 10 + 4 = 8y + 10 - 1 $\implies$ 15y - 6 = 8y + 9
You multiplied the terms outside the parentheses ( $+4$ on the left and $-1$ on the right) by the leading multipliers $5$ and $2$ . The distributive property only applies to the terms inside the parentheses: $5(3y - 2) + 4 = 15y - 10 + 4$ .
Step 2
7y = -2 $\implies$ y = - $\frac{2}{7}$
Correct
y = $\frac{19}{13}$
This step correctly isolates the variable from your previous equation, but the numerical result is affected by the distribution error in Step 1.
Step 3
$M = \frac{6\left(-\frac{2}{7}\right) - 5}{3\left(-\frac{2}{7}\right) + 1} \implies M = \frac{-\frac{12}{7} - \frac{35}{7}}{-\frac{6}{7} + \frac{7}{7}} \implies M = \frac{-\frac{47}{7}}{\frac{1}{7}}$
Correct
$M = \frac{6\left(\frac{19}{13}\right) - 5}{3\left(\frac{19}{13}\right) + 1} = \frac{\frac{49}{13}}{\frac{70}{13}}$
You substituted your calculated value of $y$ into the rational expression and accurately manipulated fractions, but the inputs were affected by the earlier error.
Step 4
M = -47
Correct
M = $\frac{7}{10}$
Your division and fraction simplification were executed properly based on your prior values, but the final numerical result is incorrect due to the early error.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly combined like terms once the initial expression was expanded.

You accurately substituted your solved value into the rational expression.

You handled complex fraction division and common denominators with precision.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Overextended Distribution

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated terms outside the parentheses as if they were enclosed inside them, multiplying the trailing constants $+4$ and $-1$ by the factors $5$ and $2$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Boundary of Distribution: In an expression of the form $a(b + c) + d$ , multiplication by $a$ applies strictly to $b$ and $c$ , leaving $d$ untouched.
Highlight or underline the closing parenthesis before expanding to clearly separate grouped terms from outside additive terms.

CORRECTED PATH

ACEpath Teacher's Demonstration

Expand only inside the parentheses: $5(3y - 2) + 4 = 15y - 10 + 4$ and $2(4y + 5) - 1 = 8y + 10 - 1$ .
Combine constant terms on each side to find $y = \frac{19}{13}$ .
Substitute $y = \frac{19}{13}$ into $M = \frac{6y - 5}{3y + 1}$ to obtain $\frac{\frac{49}{13}}{\frac{70}{13}}$ .
Simplify the fraction: $\frac{49}{70} = \frac{7}{10}$ .

Final answer

M = $\frac{7}{10}$

NEXT STEPS

Final Takeaways from Your Teacher

Practice identifying the boundaries of parenthetical expressions before distributing.
Solve linear equations with standalone constant terms on both sides without expanding past the parentheses.

Lock it in

Question

What is the algebraic rule for expanding an expression of the form $a(b - c)$ ?

Answer

$a(b - c) = ab - ac$ . The exterior factor $a$ must multiply every term inside the parentheses.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets to distribute the multiplication to the constant term inside the parentheses on the left side, obtaining 15y - 2 instead of 15y - 10.

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 forgets to reverse the sign when transposing the constant term across the equals sign, computing 9 - 6 = 3 instead of 9 + 6 = 15.

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 incorrectly distributes the multiplication across the outside terms by grouping before expanding, treating 5(3y - 2) + 4 as 5(3y - 2 + 4) or multiplying the outside term 4 by 5.

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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