Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Ask your questions and receive detailed answers from professionals with extensive experience in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
Sure, let's solve the given system of linear equations step by step using the method of elimination or substitution.
We are given the system:
[tex]\[ \begin{cases} 3x + 2y = -1 \\ 2x - 5y = 31 \end{cases} \][/tex]
### Step 1: Eliminate one of the variables
We can eliminate [tex]\( y \)[/tex] by finding common coefficients for [tex]\( y \)[/tex] in both equations, and then subtracting or adding the equations.
Let's multiply the first equation by 5 and the second equation by 2 so that the coefficients of [tex]\( y \)[/tex] in both equations are opposites:
[tex]\[ \begin{cases} 5(3x + 2y) = 5(-1) \\ 2(2x - 5y) = 2(31) \end{cases} \][/tex]
This simplifies to:
[tex]\[ \begin{cases} 15x + 10y = -5 \\ 4x - 10y = 62 \end{cases} \][/tex]
### Step 2: Add the equations to eliminate [tex]\( y \)[/tex]
Now we add the two new equations together:
[tex]\[ (15x + 10y) + (4x - 10y) = -5 + 62 \][/tex]
Simplifying this:
[tex]\[ 15x + 4x + 10y - 10y = 57 \][/tex]
This reduces to:
[tex]\[ 19x = 57 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Divide both sides by 19 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{57}{19} \][/tex]
[tex]\[ x = 3 \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into one of the original equations
Now that we have [tex]\( x = 3 \)[/tex], substitute it back into the first original equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 3(3) + 2y = -1 \][/tex]
[tex]\[ 9 + 2y = -1 \][/tex]
Subtract 9 from both sides:
[tex]\[ 2y = -1 - 9 \][/tex]
[tex]\[ 2y = -10 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{-10}{2} \][/tex]
[tex]\[ y = -5 \][/tex]
### Solution
Thus, the solution to the system of equations is:
[tex]\[ x = 3,\, y = -5 \][/tex]
So, the solution in coordinate form is:
[tex]\[ (3, -5) \][/tex]
We are given the system:
[tex]\[ \begin{cases} 3x + 2y = -1 \\ 2x - 5y = 31 \end{cases} \][/tex]
### Step 1: Eliminate one of the variables
We can eliminate [tex]\( y \)[/tex] by finding common coefficients for [tex]\( y \)[/tex] in both equations, and then subtracting or adding the equations.
Let's multiply the first equation by 5 and the second equation by 2 so that the coefficients of [tex]\( y \)[/tex] in both equations are opposites:
[tex]\[ \begin{cases} 5(3x + 2y) = 5(-1) \\ 2(2x - 5y) = 2(31) \end{cases} \][/tex]
This simplifies to:
[tex]\[ \begin{cases} 15x + 10y = -5 \\ 4x - 10y = 62 \end{cases} \][/tex]
### Step 2: Add the equations to eliminate [tex]\( y \)[/tex]
Now we add the two new equations together:
[tex]\[ (15x + 10y) + (4x - 10y) = -5 + 62 \][/tex]
Simplifying this:
[tex]\[ 15x + 4x + 10y - 10y = 57 \][/tex]
This reduces to:
[tex]\[ 19x = 57 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Divide both sides by 19 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{57}{19} \][/tex]
[tex]\[ x = 3 \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into one of the original equations
Now that we have [tex]\( x = 3 \)[/tex], substitute it back into the first original equation to solve for [tex]\( y \)[/tex]:
[tex]\[ 3(3) + 2y = -1 \][/tex]
[tex]\[ 9 + 2y = -1 \][/tex]
Subtract 9 from both sides:
[tex]\[ 2y = -1 - 9 \][/tex]
[tex]\[ 2y = -10 \][/tex]
Divide both sides by 2:
[tex]\[ y = \frac{-10}{2} \][/tex]
[tex]\[ y = -5 \][/tex]
### Solution
Thus, the solution to the system of equations is:
[tex]\[ x = 3,\, y = -5 \][/tex]
So, the solution in coordinate form is:
[tex]\[ (3, -5) \][/tex]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.