Solve the System of Equations: -8x + 3y = 7 and 24x + y = 3

Solve the above set of equations and choose the correct answer.

$\begin{cases} -8x+3y=7 \\ 24x+y=3 \end{cases}$

We will solve the system of equations using the elimination method.

Step 1: We have the system of equations:

• Equation 1: $-8x + 3y = 7$
• Equation 2: $24x + y = 3$

Step 2: Let's eliminate $x$ by aligning coefficients. Multiply Equation 1 by 3:

Equation 1: $-8x + 3y = 7$ becomes $-24x + 9y = 21$

Now subtract Equation 2 from the modified Equation 1:

$-24x + 9y - (24x + y) = 21 - 3$

Simplifying, we get:

$-48x + 8y = 18$

Notice, this was incorrect since subtraction led to an error in understanding coefficients. Let's find $y$ directly.

We have:

• Equation 1: $-8x + 3y = 7$
• Equation 2: $24x + y = 3$

Step 3: Solve for $y$ from Equation 2:

Multiply Equation 2 by 3:

$24x + y = 3$

3 $(24x + y = 3)$ gives:

$72x + 3y = 9$

Subtracting Equation 1 from this new Equation gives:

$(72x + 3y) - (-8x + 3y) = 9 - 7$

$80x = 2$

Step 4: Solve for $x$:

$x = \frac{2}{80} = 0.025$

Step 5: Substitute $x = 0.025$ back into Equation 2 to find $y$:

$24(0.025) + y = 3$

$0.6 + y = 3$

$y = 3 - 0.6 = 2.4$

Thus, the solution to the system of equations is $x = 0.025$ and $y = 2.4$.

The choice corresponding to this solution is:

$x = 0.025, y = 2.4$