Solve the system of linear equations given below using substitution or elimination.
3x+2y=12
−x+y=11
We can solve this system of linear equations using substitution or elimination.
Method 1: Substitution
Step 1: Solve one equation for one variable in terms of the other variable.
From the second equation, we have: -x + y = 11
Rearranging the equation gives:
y = x + 11
Step 2: Substitute the value of y in terms of x into the other equation.
Plugging y = x + 11 into the first equation, we have:
3x + 2(x + 11) = 12
Simplifying, we get:
3x + 2x + 22 = 12
Combining like terms, we obtain:
5x + 22 = 12
Subtracting 22 from both sides gives:
5x = -10
Dividing both sides by 5, we have:
x = -2
Step 3: Substitute the value of x back into one of the original equations to solve for the other variable.
Using the second equation:
-y = -2 + 11
Simplifying:
-y = 9
Dividing both sides by -1, we find:
y = -9
Therefore, the solution to the system of equations is x = -2 and y = -9.
Method 2: Elimination
Step 1: Multiply the second equation by 2 to make the coefficients of x match.
The second equation -x + y = 11 becomes:
-2x + 2y = 22
Step 2: Add the two equations together to eliminate the y-term.
(3x + 2y) + (-2x + 2y) = 12 + 22
Simplifying gives:
x = 34
Step 3: Substitute the value of x back into one of the original equations to solve for the other variable.
Using the first equation, we have:
3(34) + 2y = 12
Simplifying:
102 + 2y = 12
Subtracting 102 from both sides yields:
2y = -90
Dividing both sides by 2 provides:
y = -45
Thus, the solution to the system of equations is x = 34 and y = -45.
Both methods give the same solution: x = -2, y = -9.
To solve this system of linear equations using substitution or elimination, we'll start by using elimination to eliminate one variable.
Let's begin by multiplying both sides of the second equation by 3 to make the coefficients of x in both equations equal:
3*(-x + y) = 3*11
-3x + 3y = 33
Now, we have the following system of equations:
3x + 2y = 12
-3x + 3y = 33
To eliminate x, we'll add the two equations together:
(3x + 2y) + (-3x + 3y) = 12 + 33
5y = 45
Next, we can solve for y by dividing both sides of the equation by 5:
5y/5 = 45/5
y = 9
Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the second equation:
-x + y = 11
-x + 9 = 11
To isolate x, we'll subtract 9 from both sides of the equation:
-x + 9 - 9 = 11 - 9
-x = 2
Finally, we can solve for x by multiplying both sides of the equation by -1:
(-1)(-x) = -1(2)
x = -2
Therefore, the solution to the system of linear equations is x = -2 and y = 9.
To solve the system of linear equations, we can use either substitution or elimination method. Let's use the substitution method for this example.
Step 1: Solve one equation for one variable in terms of the other variable.
Let's solve the second equation for x in terms of y:
-x + y = 11
x = y + 11
Step 2: Substitute this expression for x into the other equation.
Now, substitute the value of x (y + 11) into the first equation:
3x + 2y = 12
3(y + 11) + 2y = 12
Step 3: Simplify and solve for y.
Distribute:
3y + 33 + 2y = 12
Combine like terms:
5y + 33 = 12
Subtract 33 from both sides:
5y = 12 -33
5y = -21
Divide by 5:
y = -21/5
Step 4: Substitute the value of y into either of the original equations to find x.
Let's substitute y = -21/5 into the second equation:
-x + y = 11
-x + (-21/5) = 11
Multiply through by 5 to remove the fraction:
-5x - 21 = 55
Add 21 to both sides:
-21 - 21 - 21 = 55 + 21
-5x = 76
Divide by -5:
x = 76 / -5
Now we have found the values of x and y. The solution to the system of linear equations is:
x = 76 / -5
y = -21/5