Explanation:
How to solve a system of equations using substitution:
1. Choose one of the equations and solve for one of the variables in terms of the other variable.
2. Substitute the expression you found in step 1 into the other equation for the variable you solved for.
3. Solve for the remaining variable.
4. Substitute the value you found in step 3 into either of the original equations to find the value of the other variable.
5. Check your solution by plugging in the values you found in both original equations to make sure they are true.
Example:
Solve the system of equations below using substitution:
2x + y = 8
x - y = 2
1. Solve the second equation for x in terms of y:
x = y + 2
2. Substitute the expression for x from step 1 into the first equation:
2(y + 2) + y = 8
3. Simplify and solve for y:
2y + 4 + y = 8
3y = 4
y = 4/3
4. Substitute the value of y into either of the original equations to find x:
x - (4/3) = 2
x = 8/3
5. Check the solution by plugging in both values into both original equations:
2(8/3) + (4/3) = 8 (true)
(8/3) - (4/3) = 2 (true)
Therefore, the solution to the system of equations is (8/3, 4/3).
How to solve a system of equations by graphing:
1. Rewrite each equation in slope-intercept form, y = mx + b.
2. Graph each equation on the same coordinate plane.
3. Find the point of intersection of the two lines. This is the solution to the system of equations.
4. Check your solution by plugging in the values into both original equations to make sure they are true.
Example:
Solve the system of equations below by graphing:
y = 2x - 1
y = -x + 3
1. Rewrite each equation in slope-intercept form:
y = 2x - 1 is already in slope-intercept form.
y = -x + 3 can be rewritten as y = -1x + 3.
2. Graph each equation on the same coordinate plane:
The two lines intersect at the point (1, 1).
3. Check the solution by plugging in both values into both original equations:
y = 2x - 1: 1 = 2(1) - 1 (true)
y = -x + 3: 1 = -(1) + 3 (true)
Therefore, the solution to the system of equations is (1, 1).