Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
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]
We hope this was helpful. Please come back whenever you need more information or answers to your queries. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
