Question 441·Medium·Algebra

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In the equation $7(3y - 2) = 5(y + 6) + 2(3y - 2)$ , what is the value of $3y - 2$ ?

A linear equation can have the unknown in several places, including inside a repeated expression. You can enter both sides in Desmos without distributing, or keep that repeated expression together to shorten the algebra by hand. Find the unknown, then evaluate the expression the question asks for. Stopping at the unknown is the tempting trap.

Hints

Hint 1
A linear equation can go into Desmos without expanding its parentheses. Replace every $y$ with $x_1$ and change $=$ to $\sim$ so Desmos finds the value that makes both sides equal.
Hint 2
The solution gives you $y$ , but the question asks for $3y-2$ . Once Desmos reports the value of $x_1$ , what expression should you type on the next line?

Step-by-step

Approach 1: Solve in Desmos, then evaluate
Step 1
Find the value of y
Type $7(3x_1-2)\sim5(x_1+6)+2(3x_1-2)$ . Replace every $y$ with $x_1$ ; the $\sim$ tells Desmos to fit a value that makes both sides equal. Under PARAMETERS, Desmos shows $x_1=4$ , so $y=4$ .
Step 2
Evaluate what the question asks for
The question asks for the whole expression $3y-2$ , not $y$ . Type $3x_1-2$ below the regression. Desmos uses $x_1=4$ and prints $10$ , so the value of $3y-2$ is $10$ . Choice C.

Approach 2: Keep the repeated expression together
Step 1
Remove the matching term
The expression $3y-2$ appears unchanged on both sides. Subtract $2(3y-2)$ from both sides so you can cancel that whole term without distributing: $5(3y-2)=5(y+6)$ .
Step 2
Remove the shared factor
Divide both sides by $5$ : $3y-2=y+6$ . Now the requested expression is by itself on the left.
Step 3
Collect the y terms
Subtract $y$ from both sides: $2y-2=6$ .
Step 4
Isolate the y term
Add $2$ to both sides: $2y=8$ .
Step 5
Find y
Divide both sides by $2$ : $y=4$ . That solves the equation, but it isn't yet the value requested.
Step 6
Find the requested value
Put $y=4$ into the requested expression, then multiply and subtract: $\begin{aligned}3y-2&=3(4)-2\\&=12-2\\&=10.\end{aligned}$ Choice C.

Question 441·Medium·Algebra

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Step-by-step

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Strategy

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Step-by-step

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