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

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 135 | Free SAT Prep | Aniko

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Question 135·Hard·Systems of Two Linear Equations in Two Variables

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\frac{3}{5}x + \frac{1}{2}y = -\frac{4}{5}

\frac{2}{3}x - \frac{3}{4}y = -\frac{7}{2}

The ordered pair $(x, y)$ satisfies the system of equations above. What is the value of $-2x + 5y$ ?

Your answer

(Express the answer as an integer)

For SAT system-of-equations questions with fractions, first clear all denominators to turn the system into one with integer coefficients—this makes elimination or substitution much less error-prone. Then choose the variable that can be eliminated with the smallest multipliers, solve for one variable, back-substitute to find the other, and only at the end substitute into the specific expression the question asks for (like $-2x+5y$ ), being careful with negative signs so you don’t accidentally evaluate a different expression such as $2x+5y$ or $-(2x+5y)$ .

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Make the equations easier to work with

Before trying to solve, focus on getting rid of the fractions by multiplying each equation by a suitable number so all coefficients become integers.

Choose a method to solve the system

After clearing fractions, you will have two equations with $x$ and $y$ . Think about whether it is easier to eliminate $x$ or $y$ by adding/subtracting multiples of the equations.

Use your solution to answer the actual question

Once you find $x$ and $y$ , remember that the problem asks for $-2x + 5y$ , not just $x$ or $y$ . Substitute both values into that expression.

Desmos Guide

Graph the two equations

Enter the two equations into Desmos exactly as written:

(3/5)x + (1/2)y = -4/5
(2/3)x - (3/4)y = -7/2 Desmos will plot two lines on the same coordinate plane.

Find the intersection point

Click on the point where the two lines intersect. Desmos will display its coordinates (x, y); this is the solution to the system.

Evaluate -2x + 5y using the intersection coordinates

Using the $x$ - and $y$ -values from the intersection, type a new expression in Desmos of the form -2*(x-value) + 5*(y-value). The numerical output that Desmos shows for this expression is the value of $-2x + 5y$ for the solution of the system.

Step-by-step Explanation

Clear the fractions in both equations

Start by removing fractions so the equations are easier to work with.

In the first equation, the denominators are 5 and 2. The least common multiple (LCM) is 10. Multiply the entire first equation by 10:
- $10\cdot\frac{3}{5}x + 10\cdot\frac{1}{2}y = 10\cdot\left(-\frac{4}{5}\right)$ gives $6x + 5y = -8$ .
In the second equation, the denominators are 3 and 4. The LCM is 12. Multiply the entire second equation by 12:
- $12\cdot\frac{2}{3}x - 12\cdot\frac{3}{4}y = 12\cdot\left(-\frac{7}{2}\right)$ gives $8x - 9y = -42$ .

Now the system is:

$6x + 5y = -8$
$8x - 9y = -42$ .

Use elimination to solve for y

Use elimination on the system

$6x + 5y = -8$
$8x - 9y = -42$ .

Eliminate $x$ by making the $x$ -coefficients opposites:

Multiply the first equation by 4: $24x + 20y = -32$ .
Multiply the second equation by $-3$ : $-24x + 27y = 126$ .

Add these new equations:

$24x + 20y + (-24x + 27y) = -32 + 126$
The $x$ terms cancel, giving $47y = 94$ , so $y = \dfrac{94}{47} = 2$ .

Substitute to find x

Substitute $y = 2$ into one of the simpler equations, for example $6x + 5y = -8$ :

$6x + 5(2) = -8$
$6x + 10 = -8$
$6x = -18$
$x = -3$ .

So the solution to the system is $(x, y) = (-3, 2)$ .

Evaluate the expression -2x + 5y

Now plug $x = -3$ and $y = 2$ into the expression $-2x + 5y$ :

-2(-3) + 5(2) = 6 + 10 = 16

Therefore, the value of $-2x + 5y$ is $\boxed{16}$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation