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

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SAT Systems of Two Linear Equations Question 51 | Free SAT Prep | Aniko

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Question 51·Hard·Systems of Two Linear Equations in Two Variables

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5\bigl(3x + 2y\bigr) - 4 = 2\bigl(x - 5y\bigr) \\[0.7em] 7\bigl(x - y\bigr) + 3 = 4\bigl(2x + y\bigr) - 1

The ordered pair $(x, y)$ satisfies the system of equations above. What is the value of $205(x + y)$ ?

Your answer

(Express the answer as an integer)

For systems of two linear equations, first distribute and simplify both equations into the form $ax + by = c$ . Then choose elimination or substitution—elimination is often faster when the coefficients line up nicely. Pay attention to what the question actually asks for: if it wants a combination like $x + y$ or $3x - 2y$ , you can sometimes solve directly for that combination without fully solving for both variables, or at least stop your work as soon as that combination is known. Always use clean fraction arithmetic and check signs carefully to avoid losing points on small algebra errors.

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Simplify each equation first

Before trying to solve the system, distribute the numbers outside the parentheses in both equations and combine like terms to get simpler linear equations in $x$ and $y$ .

Use elimination after simplifying

Once you have two simpler equations in the form $ax + by = c$ , think about subtracting one equation from the other to eliminate either $x$ or $y$ and get a single equation in one variable.

Relate x and y, then solve

From the equation you get after elimination, solve for one variable in terms of the other, then substitute into one of the original simplified equations to find the exact value of $y$ (and then $x$ ).

Answer only asks for a combination of x and y

You do not need $x$ and $y$ separately in the final answer—only $x + y$ . Once you know $x$ and $y$ (or directly $x + y$ ), multiply that sum by 205 to get the final value.

Desmos Guide

Enter the simplified equations

In Desmos, type the two simplified equations on separate lines: 13x + 20y = 4 and x + 11y = 4. Desmos will graph each as a line.

Find the intersection point

Zoom or pan until you see where the two lines cross. Tap or click on the intersection point; Desmos will display the coordinates $(x, y)$ of that point.

Compute x + y and then 205(x + y)

In a new expression line, type x1 + y1 using the exact coordinates from the intersection (or directly type the numeric $x$ and $y$ values you found). Then, on the next line, type 205 * (x1 + y1) (or 205 * (x + y) with your numbers) and read off the resulting value from Desmos. That value is the answer.

Step-by-step Explanation

Distribute and simplify both equations

Expand the parentheses in each equation and combine like terms.

First equation:

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

Distribute:

15x + 10y - 4 = 2x - 10y

Move all terms to one side:

15x + 10y - 4 - 2x + 10y = 0 \\[0.7em] 13x + 20y - 4 = 0 \\[0.7em] 13x + 20y = 4

Second equation:

7(x - y) + 3 = 4(2x + y) - 1

Distribute:

7x - 7y + 3 = 8x + 4y - 1

Move all terms to one side:

7x - 7y + 3 - 8x - 4y + 1 = 0 \\[0.7em] -x - 11y + 4 = 0 \\[0.7em] x + 11y = 4

So the simplified system is:

13x + 20y = 4 \\[0.7em] x + 11y = 4

Eliminate one variable using the two simplified equations

Use elimination to relate $x$ and $y$ .

Subtract the second equation from the first to eliminate $x$ :

(13x + 20y) - (x + 11y) = 4 - 4

This gives:

12x + 9y = 0

Divide everything by 3 to simplify:

4x + 3y = 0

Now you have a simpler relationship between $x$ and $y$ . You can solve this for one variable in terms of the other.

Express one variable in terms of the other

From $4x + 3y = 0$ , solve for $x$ in terms of $y$ :

4x = -3y \\[0.7em] x = -\frac{3}{4}y

Now substitute this expression for $x$ into one of the original simplified equations, such as $x + 11y = 4$ . This will let you solve for $y$ .

Solve for y

Substitute $x = -\frac{3}{4}y$ into $x + 11y = 4$ :

-\frac{3}{4}y + 11y = 4

Write $11$ as $\frac{44}{4}$ to combine like terms:

-\frac{3}{4}y + \frac{44}{4}y = 4 \\[0.7em] \frac{41}{4}y = 4

Multiply both sides by 4:

41y = 16 \\[0.7em] y = \frac{16}{41}

Find x + y, then compute 205(x + y)

You already have $x = -\frac{3}{4}y$ and $y = \frac{16}{41}$ .

First find $x$ :

x = -\frac{3}{4} \cdot \frac{16}{41} = -\frac{48}{164} = -\frac{12}{41}

Now compute $x + y$ :

x + y = -\frac{12}{41} + \frac{16}{41} = \frac{4}{41}

Finally, multiply by 205:

205(x + y) = 205 \cdot \frac{4}{41} = 5 \cdot 4 = 20

So the value of $205(x + y)$ is $20$ .

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Step-by-step Explanation

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