Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Join our platform to get reliable answers to your questions from a knowledgeable community of experts. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Let's solve the system of linear equations given:
1) [tex]\( 5x + 3y = 21 \)[/tex]
2) [tex]\( -2x + 4y = 2 \)[/tex]
We will solve this system using the method of substitution or elimination.
### Step-by-Step Solution:
#### Step 1: Set up the equations in matrix form
We can represent the system of linear equations with a coefficient matrix [tex]\( A \)[/tex] and a constant matrix [tex]\( B \)[/tex]:
[tex]\[ A \cdot \begin{pmatrix} x \\ y \end{pmatrix} = B \][/tex]
Where [tex]\( A = \begin{pmatrix} 5 & 3 \\ -2 & 4 \end{pmatrix} \)[/tex] and [tex]\( B = \begin{pmatrix} 21 \\ 2 \end{pmatrix} \)[/tex].
#### Step 2: Use substitution or elimination method
Using elimination:
1. Multiply Equation 1 by 2 to make the coefficients of [tex]\(x\)[/tex] in both equations equal:
[tex]\[ 2 \cdot (5x + 3y) = 2 \cdot 21 \][/tex]
This yields:
[tex]\[ 10x + 6y = 42 \quad \text{(Equation 3)} \][/tex]
2. Add Equation 2 to Equation 3:
[tex]\[ (10x + 6y) + (-2x + 4y) = 42 + 2 \][/tex]
Simplifying this, we get:
[tex]\[ (10x - 2x) + (6y + 4y) = 44 \][/tex]
Which simplifies to:
[tex]\[ 8x + 10y = 44 \quad \text{(Equation 4)} \][/tex]
3. Now solve Equation 4 for [tex]\( y \)[/tex]:
[tex]\[ 8x + 10y = 44 \][/tex]
Notice there might have been a likely approach mistake, instead, we normalize the equation first if needed. We can combine the coefficients accordingly instead.
Combine coefficients properly:
- Equation after normalization from the 4th:
Multiply Equation 2 by 5:
[tex]\[ 5 \cdot (-2x + 4y) = 5 \cdot 2 \][/tex]
Yielding:
[tex]\[ - 10x + 20y = 10 \quad (Equation 5) \][/tex]
Add results of valid Equation3 and Equation5:
10x+6y+(-10x+20y)=42+10
Simplifies to
26 y = 52
Recovered fast approach back. Valid substitution indeed says:
[tex]\[ y= 2 \][/tex]
#### Step 3: Solve for [tex]\(x\)[/tex]:
Using [tex]\(y=2\)[/tex] in first simpler equation
we combine coefficients as:
[tex]\[ 5x + 3(2)=21\][/tex]
Solving:
5x +6=21
\]
So
\[
5x=15
obtaining result:
\[.
Thus the results through good solving steps, recover accurate results:
\boxed{ x = 3, y=2 }
Thus:
\boxed{(3,2)}
1) [tex]\( 5x + 3y = 21 \)[/tex]
2) [tex]\( -2x + 4y = 2 \)[/tex]
We will solve this system using the method of substitution or elimination.
### Step-by-Step Solution:
#### Step 1: Set up the equations in matrix form
We can represent the system of linear equations with a coefficient matrix [tex]\( A \)[/tex] and a constant matrix [tex]\( B \)[/tex]:
[tex]\[ A \cdot \begin{pmatrix} x \\ y \end{pmatrix} = B \][/tex]
Where [tex]\( A = \begin{pmatrix} 5 & 3 \\ -2 & 4 \end{pmatrix} \)[/tex] and [tex]\( B = \begin{pmatrix} 21 \\ 2 \end{pmatrix} \)[/tex].
#### Step 2: Use substitution or elimination method
Using elimination:
1. Multiply Equation 1 by 2 to make the coefficients of [tex]\(x\)[/tex] in both equations equal:
[tex]\[ 2 \cdot (5x + 3y) = 2 \cdot 21 \][/tex]
This yields:
[tex]\[ 10x + 6y = 42 \quad \text{(Equation 3)} \][/tex]
2. Add Equation 2 to Equation 3:
[tex]\[ (10x + 6y) + (-2x + 4y) = 42 + 2 \][/tex]
Simplifying this, we get:
[tex]\[ (10x - 2x) + (6y + 4y) = 44 \][/tex]
Which simplifies to:
[tex]\[ 8x + 10y = 44 \quad \text{(Equation 4)} \][/tex]
3. Now solve Equation 4 for [tex]\( y \)[/tex]:
[tex]\[ 8x + 10y = 44 \][/tex]
Notice there might have been a likely approach mistake, instead, we normalize the equation first if needed. We can combine the coefficients accordingly instead.
Combine coefficients properly:
- Equation after normalization from the 4th:
Multiply Equation 2 by 5:
[tex]\[ 5 \cdot (-2x + 4y) = 5 \cdot 2 \][/tex]
Yielding:
[tex]\[ - 10x + 20y = 10 \quad (Equation 5) \][/tex]
Add results of valid Equation3 and Equation5:
10x+6y+(-10x+20y)=42+10
Simplifies to
26 y = 52
Recovered fast approach back. Valid substitution indeed says:
[tex]\[ y= 2 \][/tex]
#### Step 3: Solve for [tex]\(x\)[/tex]:
Using [tex]\(y=2\)[/tex] in first simpler equation
we combine coefficients as:
[tex]\[ 5x + 3(2)=21\][/tex]
Solving:
5x +6=21
\]
So
\[
5x=15
obtaining result:
\[.
Thus the results through good solving steps, recover accurate results:
\boxed{ x = 3, y=2 }
Thus:
\boxed{(3,2)}
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.