At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To solve the system of equations
[tex]\[ \begin{array}{l} 3x + 4y = 5 \\ 6x - y = 1 \end{array} \][/tex]
we can use the method of determinants, also known as Cramer's Rule. Here are the steps to find the solution [tex]\( (x, y) \)[/tex]:
### Step 1: Set up the equations
We start by setting up the two linear equations:
1. [tex]\( 3x + 4y = 5 \)[/tex]
2. [tex]\( 6x - y = 1 \)[/tex]
### Step 2: Formulate the coefficient matrix and the constant matrix
We identify the coefficients from the given equations:
For the equation [tex]\( 3x + 4y = 5 \)[/tex]:
- Coefficient of [tex]\( x \)[/tex] (a₁) = 3
- Coefficient of [tex]\( y \)[/tex] (b₁) = 4
- Constant term (c₁) = 5
For the equation [tex]\( 6x - y = 1 \)[/tex]:
- Coefficient of [tex]\( x \)[/tex] (a₂) = 6
- Coefficient of [tex]\( y \)[/tex] (b₂) = -1
- Constant term (c₂) = 1
### Step 3: Calculate the determinant of the coefficient matrix ([tex]\( \Delta \)[/tex])
The determinant [tex]\(\Delta\)[/tex] of the coefficient matrix is given by:
[tex]\[ \Delta = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - a_2 b_1 \][/tex]
Substituting the values:
[tex]\[ \Delta = (3 \cdot -1) - (6 \cdot 4) = -3 - 24 = -27 \][/tex]
### Step 4: Calculate the determinant for [tex]\( x \)[/tex] ([tex]\( \Delta_x \)[/tex])
The determinant [tex]\( \Delta_x \)[/tex] is obtained by replacing the first column of the coefficient matrix with the constants from the right-hand side:
[tex]\[ \Delta_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1 b_2 - c_2 b_1 \][/tex]
Substituting the values:
[tex]\[ \Delta_x = (5 \cdot -1) - (1 \cdot 4) = -5 - 4 = -9 \][/tex]
### Step 5: Calculate the determinant for [tex]\( y \)[/tex] ([tex]\( \Delta_y \)[/tex])
The determinant [tex]\( \Delta_y \)[/tex] is obtained by replacing the second column of the coefficient matrix with the constants from the right-hand side:
[tex]\[ \Delta_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1 c_2 - a_2 c_1 \][/tex]
Substituting the values:
[tex]\[ \Delta_y = (3 \cdot 1) - (6 \cdot 5) = 3 - 30 = -27 \][/tex]
### Step 6: Solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
Using Cramer's Rule:
[tex]\[ x = \frac{\Delta_x}{\Delta} \][/tex]
[tex]\[ y = \frac{\Delta_y}{\Delta} \][/tex]
Substituting the values:
[tex]\[ x = \frac{-9}{-27} = \frac{1}{3} \quad \text{or} \quad 0.3333 \][/tex]
[tex]\[ y = \frac{-27}{-27} = 1 \][/tex]
### Step 7: State the solution
The solution to the system of equations is:
[tex]\[ x = \frac{1}{3}, \quad y = 1 \][/tex]
Or, in decimal form:
[tex]\[ x \approx 0.3333, \quad y = 1 \][/tex]
Therefore, the values that simultaneously satisfy both equations are [tex]\( x \approx 0.3333 \)[/tex] and [tex]\( y = 1 \)[/tex].
[tex]\[ \begin{array}{l} 3x + 4y = 5 \\ 6x - y = 1 \end{array} \][/tex]
we can use the method of determinants, also known as Cramer's Rule. Here are the steps to find the solution [tex]\( (x, y) \)[/tex]:
### Step 1: Set up the equations
We start by setting up the two linear equations:
1. [tex]\( 3x + 4y = 5 \)[/tex]
2. [tex]\( 6x - y = 1 \)[/tex]
### Step 2: Formulate the coefficient matrix and the constant matrix
We identify the coefficients from the given equations:
For the equation [tex]\( 3x + 4y = 5 \)[/tex]:
- Coefficient of [tex]\( x \)[/tex] (a₁) = 3
- Coefficient of [tex]\( y \)[/tex] (b₁) = 4
- Constant term (c₁) = 5
For the equation [tex]\( 6x - y = 1 \)[/tex]:
- Coefficient of [tex]\( x \)[/tex] (a₂) = 6
- Coefficient of [tex]\( y \)[/tex] (b₂) = -1
- Constant term (c₂) = 1
### Step 3: Calculate the determinant of the coefficient matrix ([tex]\( \Delta \)[/tex])
The determinant [tex]\(\Delta\)[/tex] of the coefficient matrix is given by:
[tex]\[ \Delta = \begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix} = a_1 b_2 - a_2 b_1 \][/tex]
Substituting the values:
[tex]\[ \Delta = (3 \cdot -1) - (6 \cdot 4) = -3 - 24 = -27 \][/tex]
### Step 4: Calculate the determinant for [tex]\( x \)[/tex] ([tex]\( \Delta_x \)[/tex])
The determinant [tex]\( \Delta_x \)[/tex] is obtained by replacing the first column of the coefficient matrix with the constants from the right-hand side:
[tex]\[ \Delta_x = \begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix} = c_1 b_2 - c_2 b_1 \][/tex]
Substituting the values:
[tex]\[ \Delta_x = (5 \cdot -1) - (1 \cdot 4) = -5 - 4 = -9 \][/tex]
### Step 5: Calculate the determinant for [tex]\( y \)[/tex] ([tex]\( \Delta_y \)[/tex])
The determinant [tex]\( \Delta_y \)[/tex] is obtained by replacing the second column of the coefficient matrix with the constants from the right-hand side:
[tex]\[ \Delta_y = \begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix} = a_1 c_2 - a_2 c_1 \][/tex]
Substituting the values:
[tex]\[ \Delta_y = (3 \cdot 1) - (6 \cdot 5) = 3 - 30 = -27 \][/tex]
### Step 6: Solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
Using Cramer's Rule:
[tex]\[ x = \frac{\Delta_x}{\Delta} \][/tex]
[tex]\[ y = \frac{\Delta_y}{\Delta} \][/tex]
Substituting the values:
[tex]\[ x = \frac{-9}{-27} = \frac{1}{3} \quad \text{or} \quad 0.3333 \][/tex]
[tex]\[ y = \frac{-27}{-27} = 1 \][/tex]
### Step 7: State the solution
The solution to the system of equations is:
[tex]\[ x = \frac{1}{3}, \quad y = 1 \][/tex]
Or, in decimal form:
[tex]\[ x \approx 0.3333, \quad y = 1 \][/tex]
Therefore, the values that simultaneously satisfy both equations are [tex]\( x \approx 0.3333 \)[/tex] and [tex]\( y = 1 \)[/tex].
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
