Linear Equations in Two Variables

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.