Question 276·Medium·Algebra

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If $6(k - 2) - 5 = 5(k - 2) + 6$ , what is the value of $k - 2$ ?

A repeated piece in an equation can be treated as one unknown. Use substitution, a temporary letter for that piece, then solve the shorter equation with a Desmos regression. Find the value the question asks for, not necessarily the original variable. Expanding the parentheses first adds work and makes it easier to answer for the wrong quantity.

Hints

Hint 1
The same expression, $k-2$ , appears on both sides and is also what you need to find. Call that whole expression $u$ . What equation do you get when you replace both copies with $u$ ?
Hint 2
Your shorter equation has one unknown, $u$ . To solve it in Desmos, type $u_1$ for that unknown and use $\sim$ instead of $=$ . The value Desmos finds for $u_1$ is the value of the whole expression.

Step-by-step

Approach 1: Solve for the repeated expression
Step 1
Replace the repeated expression
The whole expression $k-2$ appears unchanged on both sides, and it’s what the question asks for. Let $u=k-2$ , so replacing each copy gives
$6u-5=5u+6.$
This substitution lets you find $k-2$ without finding $k$ first.
Step 2
Solve the shorter equation
Type $6u_1-5\sim5u_1+6$ in Desmos. The $\sim$ asks Desmos to fit the unknown $u_1$ ; it shows $u_1=11$ under PARAMETERS. Since $u$ stands for $k-2$ , the requested value is $k-2=11$ . Choice C.

Approach 2: Cancel matching copies by hand
Step 1
Leave one copy of the expression
Subtract $5(k-2)$ from both sides so the five copies on the right cancel:
$6(k-2)-5-5(k-2)=5(k-2)+6-5(k-2).$
Combine the matching copies:
$(k-2)-5=6.$
You don’t need to expand $k-2$ because the entire expression stays together.
Step 2
Find the requested expression
Add $5$ to both sides:
$k-2=11.$
That’s already the value of $k-2$ , so don’t subtract $2$ again. Choice C.

Question 276·Medium·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

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