Question 53·Medium·Systems of Two Linear Equations in Two Variables

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The system of equations below has solution $(x, y)$ .

\begin{aligned} x &= 2y - 1 \\ 3x + y &= 11 \end{aligned}

Which choice gives the value of $x$ ?

When one equation in a system is already solved for a variable, substitution is usually the fastest method. Replace that variable in the other equation with its equivalent expression, solve the resulting one-variable equation, and substitute back only if needed to find the requested variable.

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Use the equation already solved for $x$

The first equation gives $x$ directly in terms of $y$ . Substitute that expression for $x$ into the second equation.

Simplify after substituting

After replacing $x$ in the second equation, distribute the $3$ and combine the terms involving $y$ .

Return to the first equation

Once you know $y$ , use $x=2y-1$ to determine the requested value.

Desmos Guide

Rewrite the equations for graphing

Rewrite both equations in terms of $y$ :

y=\frac{x+1}{2}

and

y=11-3x

Graph the lines in Desmos

Enter y=(x+1)/2 and y=11-3x on separate expression lines in Desmos.

Read the intersection

Select the intersection point of the two lines. Its $x$ -coordinate is the requested value.

Step-by-step Explanation

Substitute for $x$

The first equation already expresses $x$ in terms of $y$ :

x = 2y - 1

Substitute $2y-1$ for $x$ in the second equation:

3(2y-1)+y=11

Solve for $y$

Simplify the substituted equation:

\begin{aligned} 3(2y-1)+y&=11 \\ 6y-3+y&=11 \\ 7y&=14 \\ y&=2 \end{aligned}

Find $x$

Substitute $y=2$ into $x=2y-1$ :

x=2(2)-1=3

Thus, the value of $x$ is $3$ , which is choice B.

Question 53·Medium·Systems of Two Linear Equations in Two Variables

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation