Consider the following system of linear equations: \begin{aligned} x + 2y &= 10 \\ 3x - y … — GRE General Test

Consider the following system of linear equations:

\begin{aligned} x + 2y &= 10 \\ 3x - y &= 9 \end{aligned}

Which of the following statements are true? Select all that apply.

The true statements are those asserting that

x = 4

, that

x + y = 7

, and that

2x + y = 11

Solving the system of linear equations yields the unique solution

x = 4

and

y = 3

. Substituting these values into the given choices demonstrates that the statements claiming

x = 4

x + y = 7

, and

2x + y = 11

are all mathematically correct.

Express

y

in terms of

x

using the second equation.

y = 3x - 9

Isolating

y

allows for simple substitution into the first equation.

Substitute

y = 3x - 9

into the first equation

x + 2y = 10

x + 2(3x - 9) = 10 \implies x + 6x - 18 = 10 \implies 7x = 28 \implies x = 4

This yields a single linear equation in terms of

x

Calculate

y

using

x = 4

y = 3(4) - 9 = 3

Substituting

x = 4

back gives the unique solution for

y

Evaluate the given statements with

(x, y) = (4, 3)

x = 4

is true;

x + y = 4 + 3 = 7

is true;

y = 4

is false (

y = 3

);

2x + y = 2(4) + 3 = 11

is true;

x - y = 4 - 3 = 1

is false.

Testing each condition determines which options are valid.

Solving a 2x2 system of linear equations using substitution to evaluate linear expressions.

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