What Value of y Makes the Equation True? Answer and Steps

Problem. What value of y makes the equation true? Here is the equation: $3(y-4)=2(y+1)$.

Answer. $y=14$ - stated at once, no waiting. Now the proof that 14 is the only one that works.

Check by substituting. Replace every $y$ with 14:

$3(14-4)=3\times10=30 \qquad 2(14+1)=2\times15=30$

Both sides land on 30, so the beam stays level. Try $y=10$: the left side gives 18, the right gives 22, and the beam tips. Guessing cannot answer what value of y makes the equation true. But this two-line check can - it is exactly what a worksheet wants when it asks what value of y makes the equation true. Answering what value of y makes the equation true becomes a two-second habit once you own it.

Behind the wording sits one old idea. A solution is a value that makes both sides equal. Ray's 1866 Algebra names the check verification: substitute the value, and if it is the true one, the two members will be equal.

The wording wears costumes - solve for y, find the solution, which number verifies it. Same job. And every version of what value of y makes the equation true runs on two routes:

1. Test values. With choices in front of you, substitute each candidate. The correct choice balances both sides - what value of y makes the equation true, found by trial.
2. Solve the equation. Always works. Ray's 1866 rule: clear fractions if needed; transpose the unknown terms to one side, the known to the other; reduce; divide by the coefficient.

Route 1 answers what value of y makes the equation true by trial; route 2 answers what value of y makes the equation true by algebra. Ray even proves a comfort (Art. 170): a simple equation has but one root. Exactly one number is out there - the answer to what value of y makes the equation true - and both routes land on it.

Problem. What value of y makes the equation true when the equation is $4(y+3)-6=30$? No choices are given - the everyday form of what value of y makes the equation true - so route 2 takes over.

Step 1, simplify. $4y+12-6=30$, so $4y+6=30$.

Step 2, transpose. $4y=24$.

Step 3, divide. $y=6$.

Verification. $4(6+3)-6=36-6=30$. Both members equal - what value of y makes the equation true, answered in one substitution.

Answer. $y=6$ - the one number that settles what value of y makes the equation true here.

Problem. What value of y makes the equation true when fractions block the way? The equation, taken from Ray's 1866 Algebra, is $3y-\frac{24-2y}{7}=y+8$.

Step 1, clear the fraction. Multiply every term by 7:

$21y-(24-2y)=7y+56$

Step 2, simplify and transpose. $21y-24+2y=7y+56$, so $23y-7y=56+24$ and $16y=80$.

Step 3, divide. $y=5$.

Verification. Left: $3\times5-\frac{24-10}{7}=15-2=13$. Right: $5+8=13$. Equal - Ray's own check.

Answer. $y=5$. Clearing fractions first kept every number honest - the Ray habit for the 1866 version of what value of y makes the equation true.

1. Answering with an expression. The question asks for a value - one number. Writing $3(y-4)$ back is not an answer.

2. Dropping a sign while transposing. From $4y+6=30$, the slip writes $4y=36$. A term that crosses the equals sign flips its sign: $4y=24$.

3. Dividing only one member. Divide both members by 4, or you break the balance the question depends on.

4. Skipping the verification. One substitution catches nearly every slip on a what value of y makes the equation true question - the cheapest insurance in algebra.

5. Checking a copied line. Substitute into the original equation, not a rewritten line. Original equations decide what value of y makes the equation true - rewrites do not. Answer what value of y makes the equation true from the original only.