Let's solve this system of equations step-by-step to determine the best estimate for its solution.
We have the following system of linear equations:
[tex]\[
\left\{\begin{array}{l}
4 x - 15 y = -43 \\
2 x - 3 y = -2
\end{array}\right.
\][/tex]
To solve these equations, let's use the method of substitution or elimination. For simplicity, we'll use the elimination method to find the values of \( x \) and \( y \).
1. Step 1: Align the equations.
[tex]\[
4 x - 15 y = -43 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
2 x - 3 y = -2 \quad \text{(Equation 2)}
\][/tex]
2. Step 2: Make the coefficients of \( x \) or \( y \) equal.
Multiply Equation 2 by 2 to make the coefficient of \( x \) equal to that in Equation 1:
[tex]\[
2 (2 x - 3 y) = 2 (-2)
\][/tex]
[tex]\[
4 x - 6 y = -4 \quad \text{(Updated Equation 2)}
\][/tex]
3. Step 3: Eliminate \( x \) by subtracting Equation 2 from Equation 1.
Subtract the updated Equation 2 from Equation 1:
[tex]\[
(4 x - 15 y) - (4 x - 6 y) = -43 - (-4)
\][/tex]
[tex]\[
4 x - 15 y - 4 x + 6 y = -43 + 4
\][/tex]
[tex]\[
-9 y = -39
\][/tex]
[tex]\[
y = \frac{-39}{-9}
\][/tex]
[tex]\[
y = \frac{39}{9}
\][/tex]
[tex]\[
y = 4.3333 \quad \text{(approximately)}
\][/tex]
4. Step 4: Substitute \( y \) back into one of the original equations to solve for \( x \).
Using Equation 2:
[tex]\[
2 x - 3(4.3333) = -2
\][/tex]
[tex]\[
2 x - 13 = -2
\][/tex]
[tex]\[
2 x = 11
\][/tex]
[tex]\[
x = \frac{11}{2}
\][/tex]
[tex]\[
x = 5.5
\][/tex]
Thus, the solution to the system of equations is approximately \( (x, y) = (5.5, 4.3333) \).
Step 5: Compare the solution with the given choices.
The choices given are:
- \( (5.5, 4.3) \)
- \( (6.5, 5.2) \)
- \( (5.1, 4.8) \)
- \( (4.8, 5.2) \)
Step 6: Identify the closest match.
The closest option to \( (5.5, 4.3333) \) is:
- \( (5.5, 4.3) \)
Therefore, the best estimate for the solution to the system is:
Best Estimate:
[tex]\((5.5, 4.3)\)[/tex]
