Q. 12 - Algebra 2

Question

Compare the quantity in Column A and the quantity in Column B. Then determine whether:

1. The quantity in Column A is greater,
2. The quantity in Column B is greater, 
3. The two quantities are equal, or
4. The relationship cannot be determined from the information given
   $\begin{array}{c}4x+3y=7\\ 2x+y=1\end{array}$
   

        $\begin{array}{cc}ColumnA& ColumnB\\ 2x+2y& 6\end{array}$

Step-by-Step Solution

Answer

The correct answer is Option C that is the two quantities are equal.

Step 1. Apply the substitution method of solving equations

The algebraic method of substitution involves solving the one of the two equations for one variable in terms of other variable and then substituting the expression so formed for the variable in the second equation.

Step 2. Solve for y in terms of x

To solve the equation $2x+y=1$ for y in terms of x, subtract 2x from both sides.

$\begin{array}{c}2x+y-2x=1-2x\\ y=1-2x\end{array}$

Step 3. Substitute the value of expression

Now, substitute $y=1-2x$ in $4x+3y=7$ to find the value of x.

$\begin{array}{c}4x+3y=7\\ 4x+3\left(1-2x\right)=7\\ 4x+3-6x=7\\ -2x=4\end{array}$

Simplify it as shown.

$\begin{array}{c}-2x=4\\ \frac{-2x}{-2}=\frac{4}{-2}\\ x=-2\end{array}$

Thus, the value of x is $- 2$.

Step 4. Substitute the value of variable

To find the value of y, substitute $x=-2$ in the equation $y=1-2x$ and then solve for as shown.

$\begin{array}{c}y=1-2x\\ y=1-2\left(-2\right)\\ y=1+4\\ y=5\end{array}$

Thus, the value of y is 5.

Step 5. Compare the quantities in Column A and Column B

The quantity in Column A is $2x+2y$. To find its value substitute the values of x and y in the expression and solve as shown below.

$\begin{array}{c}2x+2y=2\left(-2\right)+2\left(5\right)\\ =-4+10\\ =6\end{array}$

Thus, the value of the expression in Column A is 6.

It is given that quantity in Column B is also 6.

It implies that both the quantities in the two columns are equal.

Hence, Option C that is two quantities are equal is the answer of the question.