Westonci.ca is the ultimate Q&A platform, offering detailed and reliable answers from a knowledgeable community. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
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)}
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.