Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Get quick and reliable answers to your questions from a dedicated community of professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Sure, I'll walk you through the detailed, step-by-step solution for the given system of linear equations:
[tex]\[ \left\{ \begin{array}{l} 3x = 7 + y \\ 5x - 9y = 41 \end{array} \right. \][/tex]
### Step 1: Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
Starting with the first equation:
[tex]\[ 3x = 7 + y \][/tex]
Rearrange it to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the second equation
Now that we have [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], we substitute this expression into the second equation.
The second equation is:
[tex]\[ 5x - 9y = 41 \][/tex]
Substitute [tex]\( y = 3x - 7 \)[/tex]:
[tex]\[ 5x - 9(3x - 7) = 41 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, distribute the [tex]\(-9\)[/tex] through the parentheses:
[tex]\[ 5x - 27x + 63 = 41 \][/tex]
Combine like terms:
[tex]\[ -22x + 63 = 41 \][/tex]
Isolate [tex]\( x \)[/tex] by subtracting 63 from both sides:
[tex]\[ -22x = 41 - 63 \][/tex]
[tex]\[ -22x = -22 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now that we have [tex]\( x = 1 \)[/tex], substitute this value back into the expression we found for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
[tex]\[ y = 3(1) - 7 \][/tex]
[tex]\[ y = 3 - 7 \][/tex]
[tex]\[ y = -4 \][/tex]
### Conclusion
So, the solution to the system of equations is:
[tex]\[ x = 1 \][/tex]
[tex]\[ y = -4 \][/tex]
Thus, the solution to the system [tex]\(\{ (3x = 7 + y,\; 5x - 9y = 41) \}\)[/tex] is:
[tex]\[ \left( 1, -4 \right) \][/tex]
[tex]\[ \left\{ \begin{array}{l} 3x = 7 + y \\ 5x - 9y = 41 \end{array} \right. \][/tex]
### Step 1: Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]
Starting with the first equation:
[tex]\[ 3x = 7 + y \][/tex]
Rearrange it to solve for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the second equation
Now that we have [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex], we substitute this expression into the second equation.
The second equation is:
[tex]\[ 5x - 9y = 41 \][/tex]
Substitute [tex]\( y = 3x - 7 \)[/tex]:
[tex]\[ 5x - 9(3x - 7) = 41 \][/tex]
### Step 3: Simplify and solve for [tex]\( x \)[/tex]
First, distribute the [tex]\(-9\)[/tex] through the parentheses:
[tex]\[ 5x - 27x + 63 = 41 \][/tex]
Combine like terms:
[tex]\[ -22x + 63 = 41 \][/tex]
Isolate [tex]\( x \)[/tex] by subtracting 63 from both sides:
[tex]\[ -22x = 41 - 63 \][/tex]
[tex]\[ -22x = -22 \][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[ x = 1 \][/tex]
### Step 4: Solve for [tex]\( y \)[/tex]
Now that we have [tex]\( x = 1 \)[/tex], substitute this value back into the expression we found for [tex]\( y \)[/tex]:
[tex]\[ y = 3x - 7 \][/tex]
[tex]\[ y = 3(1) - 7 \][/tex]
[tex]\[ y = 3 - 7 \][/tex]
[tex]\[ y = -4 \][/tex]
### Conclusion
So, the solution to the system of equations is:
[tex]\[ x = 1 \][/tex]
[tex]\[ y = -4 \][/tex]
Thus, the solution to the system [tex]\(\{ (3x = 7 + y,\; 5x - 9y = 41) \}\)[/tex] is:
[tex]\[ \left( 1, -4 \right) \][/tex]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.