solve the system of liner equastions given below using substitution or elimination
3x+2y=12
-x+y=11
We can solve the system of linear equations using substitution.
We can start by solving the second equation for x:
-x + y = 11
x = y - 11
Now we can substitute the value of x in the first equation:
3x + 2y = 12
3(y - 11) + 2y = 12
3y - 33 + 2y = 12
5y - 33 = 12
5y = 45
y = 9
Now we can substitute the value of y back into the second equation to solve for x:
-x + 9 = 11
-x = 2
x = -2
Therefore, the solution to the system of linear equations is x = -2 and y = 9.
We will solve the given system of linear equations using the substitution method.
Step 1: Solve one of the equations for one variable in terms of the other variable. Let's solve the second equation for x:
-x + y = 11
Simplifying, we get:
x = y - 11
Step 2: Substitute the expression for x from the equation we just solved into the other equation. Substituting x = y - 11 into the first equation, we get:
3(y - 11) + 2y = 12
Step 3: Simplify and solve the resulting equation for y:
3y - 33 + 2y = 12
Combining like terms, we have:
5y - 33 = 12
Adding 33 to both sides, we get:
5y = 45
Dividing both sides by 5, we find:
y = 9
Step 4: Substitute the value of y back into one of the original equations to solve for x. Let's use the second equation:
-x + y = 11
Substituting y = 9, we have:
-x + 9 = 11
Subtracting 9 from both sides, we obtain:
-x = 2
Multiplying both sides by -1, we find:
x = -2
Therefore, the solution to the system of linear equations is x = -2 and y = 9.
To solve the system of linear equations using substitution or elimination, let's go step by step.
Method 1: Substitution
Step 1: Solve one equation for one variable.
Let's solve the second equation (-x + y = 11) for x.
Adding x to both sides, we get:
y = x + 11
Step 2: Substitute the expression for the variable into the other equation.
Now, substitute (x + 11) for y in the first equation (3x + 2y = 12).
We get:
3x + 2(x + 11) = 12
Step 3: Simplify and solve for x.
Expand the equation:
3x + 2x + 22 = 12
5x + 22 = 12
Subtract 22 from both sides:
5x = -10
Divide both sides by 5:
x = -2
Step 4: Substitute the value of x back into one of the original equations to solve for y.
Using the second equation (-x + y = 11), substitute x = -2:
-(-2) + y = 11
2 + y = 11
Subtract 2 from both sides:
y = 9
Therefore, the solution to the system of linear equations is x = -2 and y = 9.
Method 2: Elimination
Step 1: Multiply one or both equations by a suitable number to make the coefficients of one of the variables equal.
In this case, we don't need to multiply any equation.
Step 2: Add or subtract the equations to eliminate one variable.
We'll eliminate the x variable by adding the two equations together:
(3x + 2y) + (-x + y) = 12 + 11
2x + 3y = 23
Step 3: Solve the resulting equation.
Now, we have a new equation:
2x + 3y = 23
Let's solve for x:
2x = 23 - 3y
x = (23 - 3y)/2
Step 4: Substitute the expression for x back into one of the original equations to solve for y.
Using the first equation (3x + 2y = 12), substitute x = (23 - 3y)/2:
3((23 - 3y)/2) + 2y = 12
Simplify and solve for y:
(69 - 9y)/2 + 2y = 12
69 - 9y + 4y = 24
Combine like terms:
-5y = -45
Divide both sides by -5:
y = 9
Step 5: Substitute the value of y back into one of the original equations to solve for x.
Using the second equation (-x + y = 11), substitute y = 9:
-x + 9 = 11
Add x to both sides:
9 + x = 11
x = 2
Therefore, the solution to the system of linear equations is x = 2 and y = 9.
Both methods give the same solution: x = 2 and y = 9.