There are two basic ways to solve n equations with n unknowns:
substitution and elimination. Each of them can have
variations. For example, matrix methods are extensions of the
elimination method. In this section, we will explore substitution.
Substitution Method
Three variables and Three unknowns
Let the variables be x, y, and z. The equations are eq.1, eq.2,
and eq.3. The substitution method has these basic steps:
- Solve eq.1 for z.
- Substitute the whole right-hand side of the equation (the one you
just wrote) in place of "z" in eq.2. You should
use parentheses around the substituted expression to keep everything
orderly.
- Solve the equation you wrote in step 2 for y.
- Substitute the whole right-hand side of the equation you wrote in step 1
in place of "z" in eq.3. Then substitute the whole
right-hand side of the equation you wrote in step 3 in place of "y" in
eq.3.
- Solve the equation you wrote in step 4 for x. You should get a
numeric value of x.
- Use this value of x in the equation you wrote in step 3 to solve for
y. You should get a numeric value of y.
- Use these values of x and y in the equation you wrote in step 1 to solve
for z. You should get a numeric value of z.
Now I will apply these steps
to an example, and I will label them so you can understand what
I'm doing at each step.
Example: three variables and three unknowns
3x-z+120=y
y-2z=30
x+y+z=180
Step 1: Solve eq.1 for z.
3x-z+120=y
z=3x+120-y
Step 2: Substitute z in eq.2
y-2z=30
y-2(3x+120-y)=30
Step 3: Solve the equation you wrote in step 2 for y.
y-2(3x+120-y)=30
y-6x-240+2y=30
3y-6x=270
y-2x=90
y=90+2x
Step 4: Substitute z, then substitute y in eq.3.
x+y+z=180
x+y+(3x+120-y)=180
x+(90+2x)+(3x+120-(90+2x))=180
Step 5: Solve the equation you wrote in step 4 to get the value of x.
x+(90+2x)+(3x+120-(90+2x))=180
x+90+2x+3x+120-90-2x=180
4x+120=180
x=15
Step 6: Use this value of x in the equation you wrote in step 3 to solve
for y.
y=90+2x
y=90+2(15)
y=120
Step 7: Use these values of x and y in the equation you wrote in step 1 to
solve for z.
z=3x+120-y
z=3(15)+120-120
z=45+120-120
z=45
Now you should check your
answers in the original equations.
3x-z+120=y
3(15)-45+120=120
45-45+120=120
120=120
True.
y-2z=30
120-2(45)=30
120-90=30
30=30
True.
x+y+z=180
15+120+45=180
180=180
True.
You should be able to see how this generalizes to n equations in n
unknowns. You would solve the first equation for the first unknown, then
substitute this in the second equation, and solve for the second unknown.
Then substitute both variables in the third equation, and solve for the third
unknown. Then substitute all three variables in the fourth equation, and
solve for the fourth unknown, etc. When you get to the last equation, and
solve it for the last unknown, you should be down to a numeric answer.
Then work backwards through the odd-numbered steps, getting numeric answers for
each variable in turn.
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