Question 644·Hard·Algebra

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Solve the system of equations.

\begin{cases} \dfrac{3}{4}x - \dfrac{2}{5}y = 1 \\ \dfrac{1}{2}x + \dfrac{3}{5}y = 4 \end{cases}

The solution to the system is $(x, y)$ . What is the value of $13(3x - 2y)$ ?

Your answer

A linear system gives two equations that the same pair $(x,y)$ must satisfy. Desmos can show where their lines cross, but the displayed coordinates may be rounded. If the question asks for an expression rather than the pair, try scaling and combining the whole equations to build that expression exactly.

Hints

Hint 1
The intersection of two graphed lines gives a pair that satisfies both equations. Click it to see where the solution lies, but don't rely on rounded coordinates for an exact grid-in answer.
Hint 2
Distribute the $13$ across the entire requested expression. What coefficients of $x$ and $y$ would you need to build by combining the given equations?
Hint 3
You can multiply each equation by a different number, then add them. The multipliers must act on both sides. Can you choose them so the $x$ - and $y$ -coefficients match the expression you need?

Step-by-step

Build the target from the equations
Step 1
Graph the system
Type each equation as written, then click where the lines cross. Desmos shows about $(3.385,3.846)$ . That intersection satisfies both equations, but its rounded coordinates aren't enough for an exact grid-in answer.
Step 2
Identify the coefficients you need
Distribute $13$ across the whole expression:
$13(3x-2y)=39x-26y.$
Step 3
Find multipliers for the equations
Call the first equation's multiplier $a$ and the second's $b$ . Match the coefficients of the requested expression, not the coordinates. Type $\left[\frac34a+\frac12b,-\frac25a+\frac35b\right]\sim[39,-26]$ . The two list entries match the $x$ - and $y$ -coefficients in that order. The $\sim$ asks Desmos to find $a$ and $b$ ; under PARAMETERS, it shows $a=56$ and $b=-6$ .
Step 4
Combine both whole equations
Multiply the first equation by $56$ and the second by $-6$ , then add. Scale both sides, including the $1$ and the $4$ :
$56\left(\frac34x-\frac25y\right)-6\left(\frac12x+\frac35y\right)=56(1)-6(4).$
Step 5
Recognize the requested expression
Collect the terms on the left. The multipliers were chosen to give exactly the coefficients from step 2:
$39x-26y=56(1)-6(4).$
You don't need to find either coordinate exactly.
Step 6
Evaluate the right side
Type $56(1)-6(4)$ into Desmos. It prints $32$ . Since $39x-26y$ is $13(3x-2y)$ , the exact value requested is $32$ . Grid in 32.

Question 644·Hard·Algebra

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

Hints

Step-by-step

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