Solve (y-5)-6(3+y)=7: Step-by-Step Linear Equation

Q: Why does -6 times y give me -6y and not +6y?

When you have -6(3+y), the negative 6 multiplies everything inside the parentheses. So -6 × y = -6y, not +6y. Think of it as: -6(3+y) = (-6)(3) + (-6)(y) = -18 + (-6y).

Q: How do I know which terms are like terms?

Like terms have the same variable with the same power. In this problem, y and -6y are like terms (both have y¹), while -5 and -18 are like terms (both are constants with no variables).

Q: How can I check if y = -6 is really correct?

Substitute y = -6 into the original equation: (-6-5) - 6(3+(-6)) = 7(-11) - 6(-3) = 7-11 + 18 = 7 ✓Since 7 = 7, your answer is correct!

Q: Why do we add 23 to both sides instead of subtracting?

We have -5y - 23 = 7. To eliminate the -23, we need to do the opposite operation. The opposite of subtracting 23 is adding 23, so we add 23 to both sides to get -5y = 30.

Linear Equations with Distribution and Negative Coefficients

$(y-5)-6(3+y)=7$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:03 Open parentheses properly, multiply by each factor

00:10 Collect like terms

00:19 Arrange the equation so that one side has only the unknown Y

00:29 Isolate Y

00:34 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$(y-5)-6(3+y)=7$

Step-by-step solution

Let's solve the given linear equation step-by-step:

Step 1: Distribute the $-6$ across the terms inside the parenthesis:
The given equation is $(y - 5) - 6(3 + y) = 7$ .
Distribute $-6$ to both $3$ and $y$ :
$-6 \times 3 = -18$ and $-6 \times y = -6y$ .
Rewrite the equation as: $y - 5 - 18 - 6y = 7$ .
Step 2: Combine like terms:
Combine $y$ and $-6y$ to get $-5y$ .
Combine $-5$ and $-18$ to get $-23$ .
This simplifies the equation to: $-5y - 23 = 7$ .
Step 3: Isolate the variable $y$ :
Add $23$ to both sides to eliminate $-23$ :
$-5y - 23 + 23 = 7 + 23$ simplifies to $-5y = 30$ .
Divide both sides by $-5$ to solve for $y$ :
$y = \frac{30}{-5} = -6$ .

Thus, the solution to the equation is $y = -6$ .

Final Answer

$-6$

Key Points to Remember

Essential concepts to master this topic

Distribution: Apply distributive property to terms inside parentheses first
Technique: Combine like terms: $y - 6y = -5y$ and $-5 - 18 = -23$
Check: Substitute $y = -6$ : $(-6-5) - 6(3+(-6)) = 7$ ✓

Common Mistakes

Avoid these frequent errors

Incorrect distribution of negative coefficients
Don't distribute -6 as -6(3+y) = -18+6y! This gives a positive 6y instead of negative. The negative sign affects both terms inside parentheses. Always distribute carefully: -6(3+y) = -18-6y.

Practice Quiz

Test your knowledge with interactive questions

Solve for x:

$$ 2(4-x)=8 $$

FAQ

Everything you need to know about this question

Why does -6 times y give me -6y and not +6y?

When you have -6(3+y), the negative 6 multiplies everything inside the parentheses. So -6 × y = -6y, not +6y. Think of it as: -6(3+y) = (-6)(3) + (-6)(y) = -18 + (-6y).

How do I know which terms are like terms?

Like terms have the same variable with the same power. In this problem, y and -6y are like terms (both have y¹), while -5 and -18 are like terms (both are constants with no variables).

What if I get confused with all the negative signs?

Take it step by step! Write each operation clearly:

Start: (y-5) - 6(3+y) = 7
Distribute: y - 5 - 18 - 6y = 7
Group: (y - 6y) + (-5 - 18) = 7
Simplify: -5y - 23 = 7

How can I check if y = -6 is really correct?

Substitute y = -6 into the original equation:
(-6-5) - 6(3+(-6)) = 7
(-11) - 6(-3) = 7
-11 + 18 = 7 ✓
Since 7 = 7, your answer is correct!

Why do we add 23 to both sides instead of subtracting?

We have -5y - 23 = 7. To eliminate the -23, we need to do the opposite operation. The opposite of subtracting 23 is adding 23, so we add 23 to both sides to get -5y = 30.

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