Linear Equations in Two Variables

1. Which ordered pair is a solution of the equation?

\(y=7 x-3\)

Choose 1 answer:

A. Only \((1, 4) \)

B. Only \((-1, -4) \)

C. Both \((1, 4) \) and \((-1, -4) \)

D. Neither

2. Which ordered pair is a solution of the equation?

\(y=-2 x+5\)

A. Only \((2, -9) \)

B. Only \((-2, 9) \)

C. Both \((2, -9) \) and \((-2, 9) \)

D. Neither

3. Which ordered pair is a solution of the equation?

\(2 x+4 y=6 x-y\)

A. Only \((4, 5) \)

B. Only \((5, 4) \)

C. Both \((4, 5) \) and \((5, 4) \)

D. Neither

4. Which ordered pair is a solution of the equation?

\(-x-4 y=-10\)

A. Only \((3, 2) \)

B. Only \((-3, 3) \)

C. Both \((3, 2) \) and \((-3, 3) \)

D. Neither

1. A. Only \((1, 4) \)

To check whether an ordered pair \((a,b)\) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

To check whether \((1,4)\) is a solution of the equation, let's substitute \(x=1\) and \(y=4\) into the equation:

\(\begin{aligned}

&y=7 x-3 \\

&4=7 \cdot 1-3 \\

&4=7-3 \\

&4=4

\end{aligned}\)

Since \(4=4\), we obtained a true statement, so \((1, 4)\) is indeed a solution of the equation.

To check whether \((-1, -4)\) is a solution of the equation, let's substitute \(x=-1\) and \(y=-4\) into the equation:

\(\begin{aligned}

y &=7 x-3 \\

-4 &=7 \cdot(-1)-3 \\

-4 &=-7-3 \\

-4 &=-10

\end{aligned}\)

Since \(-4 \neq-10\), we obtained a false statement, so \((-1, -4)\) is not a solution of the equation.

Only \((1, 4)\) a solution of the equation.

2. B. Only \((-2, 9) \)

To check whether an ordered pair \((a, b) \) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

To check whether \( (2, -9\), let's substitute \(x=2\) and \(y=-9\) into the equation:

\(\begin{aligned}

y &=-2 x+5 \\

-9 &=-2 \cdot 2+5 \\

-9 &=-4+5 \\

-9 &=1

\end{aligned}\)

Since \( -9 \neq 1\), we obtained a false statement, so \((2, -9) \) is not a solution of the equation.

To check whether \( (-2, 9)\), let's substitute \(x=-2\) and \(y=9\) into the equation:

\(\begin{aligned}

&y=-2 x+5 \\

&9=-2 \cdot(-2)+5 \\

&9=4+5 \\

&9=9

\end{aligned}\)

Since \(9 = 9\), we obtained a true statement, so \( (-2, 9 )\) is indeed a solution of the equation.

Only \( (-2, 9 )\) is a solution of the equation.

3. B. Only \((5, 4) \)

To check whether an ordered pair \((a,b)\) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

To check whether\((4, 5) \) is a solution of the equation, let's substitute \(x=4\) and \(y=5\) into the equation:

\(\begin{aligned}

2 x+4 y &=6 x-y \\

2 \cdot 4+4 \cdot 5 &=6 \cdot 4-5 \\

8+20 &=24-5 \\

28 &=19

\end{aligned}\)

Since \(28 \neq 19\), we obtained a false statement, so \((4, 5)\) is not a solution of the equation.

To check whether\((5, 4) \) is a solution of the equation, let's substitute \(x=5\) and \(y=4\) into the equation:

\(\begin{aligned}

2 x+4 y &=6 x-y \\

2 \cdot 5+4 \cdot 4 &=6 \cdot 5-4 \\

10+16 &=30-4 \\

26 &=26

\end{aligned}\)

Since \(26 = 26\), we obtained a true statement, so \((5, 4)\) is indeed a solution of the equation.

Only \((5, 4)\)is a solution of the equation.

4. D. Neither

To check whether an ordered pair \((a, b\) is a solution of an equation, substitute these values into the equation and determine if the resulting equality is true or false.

To check whether \((3, 2) \) is a solution of the equation, let's substitute \(x= 3\) and \(y = 2\) into the equation:

\(\begin{aligned}

-x-4 y &=-10 \\

-3-4 \cdot 2 &=-10 \\

-3-8 &=-10 \\

-11 &=-10

\end{aligned}\)

Since \( -11 \neq-10\), we obtained a false statement, so \((3, 2) \) is not a solution of the equation.

To check whether \((-3, 3) \) is a solution of the equation, let's substitute \(x= -3\) and \(y = 3\) into the equation:

\( \begin{aligned}

-x-4 y &=-10 \\

-(-3)-4 \cdot 3 &=-10 \\

3-12 &=-10 \\

-9 &=-10

\end{aligned}\)

Since \(-9 \neq-10\), we obtained a false statement, so \((-3, 3) \) is not a solution of the equation.

Neither of the ordered pairs is a solution of the equation.

1. \(y-4=-2(x+3)\)

Complete the missing value in the solution to the equation.

\( (-3, \text { ____ }) \)

2.\(-4 x-y=24\)

Complete the missing value in the solution to the equation.

\( (\text { ____ }, 8) \)

3. \(-3 x+7 y=5 x+2 y\)

Complete the missing value in the solution to the equation.

\( (-5, \text { ____ }) \)

4. \(2 x+3 y=12\)

Complete the missing value in the solution to the equation.

\( (\text { ____ }, 8) \)

1. \((-3,4)\)

To find the \(y\)-value that corresponds to \(x=3\), let's substitute this \(x\)-value in the equation.

\(\begin{aligned}

&y-4=-2(x+3) \\

&y-4=-2(-3+3) \\

&y-4=-2 \cdot 0 \\

&y-4=0 \\

&y=4

\end{aligned}\)

Therefore \((-3,4)\) is a solution of the equation.

2. \( (-8, 8)\)

To find the \(x\)-value that corresponds to \(y=8\), let's substitute this \(y\)-value in the equation.

\(\begin{aligned}

-4 x-y &=24 \\

-4 x-8 &=24 \\

-32 &=4 x \\

-8 &=x

\end{aligned}\)

Therefore \( (-8, 8)\) is a solution of the equation.

3. \((-5, -8)\)

To find the \(y\)-value that corresponds to \(x = -5\), let's substitute this xxx-value in the equation.

\(\begin{aligned}

-3 x+7 y &=5 x+2 y \\

-3 \cdot(-5)+7 y &=5 \cdot(-5)+2 y \\

15+7 y &=-25+2 y \\

5 y &=-40 \\

y &=-8

\end{aligned}\)

Therefore \((-5, -8)\) is a solution of the equation.

4. \( (-6, 8) \)

To find the \(x\)-value that corresponds to \(y = 8\), let's substitute this yyy-value in the equation.

\(\begin{aligned}

2 x+3 y &=12 \\

2 x+3 \cdot 8 &=12 \\

2 x+24 &=12 \\

2 x &=-12 \\

x &=-6

\end{aligned}\)

Therefore \((-6, 8) \) is a solution of the equation.