What is the solution to the given system of equations?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 133 | Free SAT Prep | Aniko

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Question 133·Easy·Systems of Two Linear Equations in Two Variables

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$3x - y = 7$

$x + y = 5$

What is the solution $(x, y)$ to the given system of equations?

For systems of two linear equations with nice coefficients, quickly scan for an easy elimination: if adding or subtracting the equations cancels one variable (like the $y$ terms here), use that to get a one-variable equation, solve it, then substitute back into either original equation to find the other variable. Always verify that your final $(x, y)$ pair makes both original equations true; this is faster and less error-prone on the SAT than guessing or checking only one equation.

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Look for an easy elimination

Compare the two equations: $3x - y = 7$ and $x + y = 5$ . What happens to the $y$ terms if you add the equations together?

Reduce to one variable

After you combine the equations so that $y$ disappears, you should get an equation with only $x$ . How do you solve that equation for $x$ ?

Use substitution to find the second variable

Once you know $x$ , plug it into one of the original equations, such as $x + y = 5$ , and solve that equation to find $y$ .

Desmos Guide

Graph the first equation

In Desmos, rewrite the first equation as $y = 3x - 7$ and type y = 3x - 7 into the first expression line. This graphs the line for $3x - y = 7$ .

Graph the second equation

Type y = 5 - x into the next expression line. This graphs the line for $x + y = 5$ .

Find the intersection point

Look for the point where the two lines cross. Tap or click on that intersection point; Desmos will display its coordinates. Those coordinates give the solution $(x, y)$ to the system.

Step-by-step Explanation

Write the system and choose a method

We are given the system:

$3x - y = 7$
$x + y = 5$

Notice that if we add these two equations, the $y$ terms will cancel, so elimination (adding the equations) is a quick method.

Eliminate $y$ by adding the equations

Add the left sides and the right sides of the two equations:

Left side: $(3x - y) + (x + y) = 3x - y + x + y = 4x$
Right side: $7 + 5 = 12$

So the system simplifies to a single equation in one variable: $4x = 12$ .

Solve for $x$ and then for $y$ (conceptually)

From $4x = 12$ , divide both sides by $4$ to get the value of $x$ .

Then take that value of $x$ and substitute it into the simpler equation $x + y = 5$ .

Solve $x + y = 5$ for $y$ by subtracting your value of $x$ from $5$ .

State the solution as an ordered pair

Dividing $4x = 12$ by $4$ gives $x = 3$ .

Substitute into $x + y = 5$ :

$3 + y = 5$ , so $y = 2$ .

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

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation