Respuesta :
Answer:
Solve for a variable in either equation. (If possible, choose a variable that does not have a coefficient to avoid working with fractions.)
Rewrite this equation solved for x.
x + 3y = 18
x = −3y + 18
Next, substitute (−3y + 18) in for x into the other equation. Solve for y.
2( 3y + 12x + y = 11
2(−3y + 18) + y = 11-------Substitute -3y + 18 in for
−6y + 36 + y = 11-------Distribute.
2(3y −5y + 36 = 11-------Combine like terms.
2(3y + 18−5y = −25-----Subtract 36 from both sides
2(3y + 18) + y = 5-------Divide both sides by -5.
Then, substitute y = 5 into the equation that is solved for x to find x.
x = −3y + 18
x = −3(5) + 18
x = −15 + 18
x = 3
Identify the solution. A check using x = 3 and y = 5 in both equations will show that the solution is the ordered pair (3, 5).
Elimination Method
Another way to solve a system of equations is by using the elimination method. The aim of using the elimination method is to have one variable cancel out. The resulting sum will contain a single variable that can then be identified. Once one variable is found, it can be substituted into either of the original equations to find the other variable.
Example
Find the solution to the system of equations by using the elimination method.
x − 2y = 9
3x + 2y = 11
Solution
Add the equations.
x − 2y = 9
3x + 2y = 11
4x + 2y = 20
Isolate the variable in the new equation
4x = 20
x = 5
Substitute x = 5 into either of the original equations to find y.
x − 2y = 9
(5) − 2y = 9
−2y = 4
y = −2
Identify the ordered pair that is the solution. A check in both equations will show that (5, −2) is a solution.
It may be necessary to multiply one or both of the equations in the system by a constant in order to obtain a variable that can be eliminated by addition. For example, consider the system of equations below:
3x + 2y = 6
x − y = 8
Both sides of the second equation above could be multiplied by −3. Multiplying the equation by the same number on both sides does not change the value of the equation. It will result in an equation whereby the x values can be eliminated through addition.
Special Cases
In some circumstances, both variables will drop out when adding the equations. If the resulting expression is not true, then the system is inconsistent and has no solution.
4x + 6y = 13
6x + 9y = 17
3(4x + 6y = 13)
2(6x + 9y = 17)
12x + 18y = 39
12x + 18y = 34
0 = 5
The equation is false. The system has no solution.
If both variables drop out and the resulting expression is true, then the system is dependent and has infinite solutions.
6x + 15y = 24
4x + 10y = 16
2(6x + 15y = 24)
3(4x + 10y = 16)
12x + 30y = 48
12x + 30y = 48
0 = 0
The equation is true. The system has an infinite number of solutions. (Notice that both of the original equations reduce to 2x + 5y = 8. All solutions to the system lie on this line.)
Step-by-step explanation:
