Solve the system of equations. The ordered pair satisfies the system. What is the value of ?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 48 | Free SAT Prep | Aniko

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Question 48·Hard·Systems of Two Linear Equations in Two Variables

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Solve the system of equations.

\begin{aligned} 6\bigl(2x - y\bigr) - 5\bigl(3x + 4y\bigr) &= 14 \\ 4\bigl(5x + 2y\bigr) + 3\bigl(2x - y\bigr) &= 99 \end{aligned}

The ordered pair $(x, y)$ satisfies the system. What is the value of $3x + 2y$ ?

Your answer

(Express the answer as an integer)

For systems where the question asks for a specific expression like $3x + 2y$ , first simplify any parentheses so you have a clean pair of linear equations in $x$ and $y$ . Use elimination to quickly solve the system, choosing to eliminate the variable that gives smaller coefficients, and be careful with negative signs when multiplying equations. Once you have $x$ and $y$ , immediately plug them into the requested expression—do not spend extra time solving in a more complicated way to get the expression directly unless you are very comfortable with combining equations to match that expression.

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Simplify each equation first

Before trying to solve, distribute the numbers outside the parentheses in both equations so each becomes a standard linear equation in the form $ax + by = c$ .

Use a system-solving method

Once you have two simplified equations in $x$ and $y$ , choose elimination or substitution to solve the system. Aim to eliminate either $x$ or $y$ by combining the equations.

Remember what the question actually asks

After finding $x$ and $y$ , you are not done yet—plug those values into $3x + 2y$ to answer the question.

Desmos Guide

Enter the two equations

In Desmos, type the two equations exactly as given:

6(2x - y) - 5(3x + 4y) = 14
4(5x + 2y) + 3(2x - y) = 99 Desmos will graph each as a straight line.

Find the intersection point

Use the intersection tool or tap/click where the two lines cross. Desmos will display the intersection coordinates $(x, y)$ ; this is the solution to the system.

Compute 3x + 2y in Desmos

On a new line, define the intersection point as, for example, A = intersection(y1, y2) (or use whatever labels Desmos assigned), then enter 3*A.x + 2*A.y. The numeric output Desmos gives is the value of $3x + 2y$ for the solution point.

Step-by-step Explanation

Expand and simplify both equations

Start by distributing in each equation to get them into the form $ax + by = c$ .

First equation:

6(2x - y) - 5(3x + 4y) = 14

Distribute:

12x - 6y - 15x - 20y = 14

Combine like terms:

-3x - 26y = 14

Second equation:

4(5x + 2y) + 3(2x - y) = 99

Distribute:

20x + 8y + 6x - 3y = 99

Combine like terms:

26x + 5y = 99

So the simplified system is

\begin{aligned} -3x - 26y &= 14 \\ 26x + 5y &= 99 \end{aligned}

Solve the system for x and y

Use elimination to solve the system

\begin{aligned} -3x - 26y &= 14 \\ 26x + 5y &= 99. \end{aligned}

Eliminate $x$ by making the $x$ -coefficients opposites. Multiply the first equation by $26$ and the second by $3$ :

\begin{aligned} 26(-3x - 26y) &= 26 \cdot 14 \\ 3(26x + 5y) &= 3 \cdot 99 \end{aligned}

which gives

\begin{aligned} -78x - 676y &= 364 \\ 78x + 15y &= 297. \end{aligned}

Add these two equations:

(-78x + 78x) + (-676y + 15y) = 364 + 297

-661y = 661,

which gives

y = -1.

Substitute $y = -1$ into one of the simpler equations, for example $26x + 5y = 99$ :

26x + 5(-1) = 99 \Rightarrow 26x - 5 = 99 \Rightarrow 26x = 104 \Rightarrow x = 4.

Compute the requested expression 3x + 2y

Now use the found values $x = 4$ and $y = -1$ in the expression $3x + 2y$ :

3x + 2y = 3(4) + 2(-1) = 12 - 2 = 10.

So the value of $3x + 2y$ is 10.

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation