The system of equations is Which of the following points is the solution to the system in the -plane?

Solution available with step-by-step explanation

SAT Systems of Two Linear Equations Question 73 | Free SAT Prep | Aniko

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Question 73·Easy·Systems of Two Linear Equations in Two Variables

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The system of equations is

y = 2x + 1 \\[0.7em] x + y = 7

Which of the following points $(x, y)$ is the solution to the system in the $xy$ -plane?

For systems of two linear equations, look for the easiest way to eliminate one variable—often by substitution if one equation is already solved for a variable, as here with $y = 2x + 1$ . Substitute into the other equation, solve for the remaining variable, then plug back to find the second variable and write the solution as an ordered pair. On multiple-choice questions, you can also quickly check each option by plugging its $x$ and $y$ values into both equations, but always make sure the correct pair satisfies both equations, not just one.

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Choose a method

Notice that one equation is already solved for $y$ . This makes substitution a convenient method to solve the system.

Substitute into the second equation

Take the expression for $y$ from $y = 2x + 1$ and plug it into $x + y = 7$ in place of $y$ . What equation in terms of $x$ only do you get?

Solve step by step

After you substitute, combine like terms, solve for $x$ , then plug that value back into $y = 2x + 1$ to find $y$ . Match the ordered pair you get to the answer choices.

Desmos Guide

Graph the first equation

In Desmos, type the first equation exactly as given: y = 2x + 1. You will see a straight line appear.

Graph the second equation

Type the second equation: x + y = 7. Desmos can graph this implicitly, and another straight line will appear.

Find the intersection point

Zoom or move the graph if needed and click (or tap) on the point where the two lines cross. Desmos will display its coordinates $(x, y)$ ; that ordered pair is the solution to the system. Choose the answer option that matches these coordinates.

Step-by-step Explanation

Use substitution to write one equation in terms of a single variable

The first equation is already solved for $y$ :

y = 2x + 1

Use this to substitute for $y$ in the second equation $x + y = 7$ .

Replace $y$ with $2x + 1$ in the second equation:

x + (2x + 1) = 7

Now you have an equation with only $x$ .

Solve for x

Simplify and solve the equation from Step 1:

\begin{aligned} x + 2x + 1 &= 7 \\ 3x + 1 &= 7 \\ 3x &= 6 \\ x &= 2 \end{aligned}

So the $x$ -coordinate of the solution is $2$ .

Find y and identify the solution point

Now substitute $x = 2$ back into $y = 2x + 1$ to find $y$ :

y = 2(2) + 1 = 4 + 1 = 5

So the solution to the system is the point $(x, y) = (2, 5)$ , which matches answer choice D.

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