To solve a system of equations, we need to find the values of the variables that make both equations true simultaneously. Several methods for solving systems of equations include substitution, elimination, and graphing.
Substitution method:
- Solve one equation for one variable in terms of the other variable.
- Substitute the expression obtained in step 1 into the other equation in place of the variable just solved for.
- Solve the resulting equation for the remaining variable.
- Substitute the value obtained in Step 3 back into either of the original equations to find the value of the other variable.
Elimination method:
- Multiply one or both equations by constants so that the coefficients of one of the variables are equal in both equations, but have opposite signs.
- Add the resulting equations together to eliminate one of the variables.
- Solve the resulting equation for the remaining variable.
- Substitute the value obtained in Step 3 back into either of the original equations to find the value of the other variable.
Graphing method:
- Graph both equations on the same set of axes.
- The solution to the system of equations is the point(s) where the two lines intersect.
Let's solve this using substitution and then verify with the graphing method:
To solve the system of equations y = 2x - 1 and y = 5 - x, we can substitute the expression 2x - 1 for y in the second equation and solve for x:
y = 5 - x
2x - 1 = 5 - x (substituting y = 2x - 1 for y)
3x = 6
x = 2
Now that we have found x = 2, we can substitute this value into either equation to solve for y:
y = 2x - 1
y = 2(2) - 1
y = 3
We can verify this with the graphing method by looking at where the two lines intersect. Upon inspecting the graph, the lines intersect at (2,3), verifying that the algebraic method was correct. Therefore, the solution to the system of equations is (2,3)