8-8 Using Algebra to Solve Linear Systems
Caution!
When solving systems of equations,
remember to find values for all of the
variables.
8-8 Using Algebra to Solve Linear Systems
Additional Example 1A: Solving Systems of
Equations
Solve the system of equations.
y = 4x – 6
y=x+3
The expressions x + 3 and 4x – 6 both equal y.
So by the Transitive Property they are equal to
each other.
y = 4x – 6
y=x+3
4x – 6 = x + 3
8-8 Using Algebra to Solve Linear Systems
Additional Example 1A Continued
Solve the equation to find x.
4x – 6 = x + 3
–x
–x
Subtract x from both sides.
3x – 6 =
3
+6
+6
Add 6 to both sides.
3x
9
Divide both sides by 3.
3
=
3
x
=
3
To find y, substitute 3 for x in one of the original
equations.
y=x+3=3+3=6
The solution is (3, 6).
8-8 Using Algebra to Solve Linear Systems
Additional Example 1B: Solving Systems of
Equations
y = 2x + 9
y = –8 + 2x
8-8 Using Algebra to Solve Linear Systems
Check It Out: Example 1A
Solve each system of equations.
y=x–5
y = 2x – 8
8-8 Using Algebra to Solve Linear Systems
Check It Out: Example 1B
Solve each system of equations.
y = 2x
y=x+6
8-8 Using Algebra to Solve Linear Systems
When equations in a system are not
already solve for one variable, you can
solve both equations for x or both for y.
8-8 Using Algebra to Solve Linear Systems
Additional Example 2A: Solving Systems of
Equations by Solving for a Variable
Solve the system of equations.
x + 4y = –10
x + 4y = –10
–4y
–4y
x
Solve both
equations
for x.
x – 3y = 11
x – 3y = 11
+ 3y
+ 3y
= –10 – 4y
x
= 11 + 3y
–10 – 4y = 11 + 3y
–10 – 4y = 11 + 3y
– 3y
– 3y
–10 – 7y = 11
Subtract 3y
from both
sides.
8-8 Using Algebra to Solve Linear Systems
Additional Example 2A Continued
–10 – 7y = 11
+10
+10
– 7y
21
–7 = – 7
Add 10 to both sides.
Divide both sides by –7.
y = –3
x = 11 + 3y
= 11 + 3(–3) Substitute –3 for y.
= 11 + –9 = 2
The solution is (2, –3).
8-8 Using Algebra to Solve Linear Systems
Helpful Hint
You can solve for either variable. It is usually
easiest to solve for a variable that has a
coefficient of 1.
8-8 Using Algebra to Solve Linear Systems
Additional Example 2B: Solving Systems of
Equations by Solving for a Variable
Solve the system of equations.
–2x + 10y = –8
Solve both x – 5y = 4
–2x + 10y = –8
x – 5y = 4
equations
–10y
–10y for x.
+5y
+5y
–2x
= –8 – 10y
x
= 4 + 5y
–2x = –8 – 10y
–2
–2 –2
x = 4 + 5y
4 + 5y = 4 + 5y Subtract 5y
– 5y
– 5y from both
sides.
4=4
Since 4 = 4 is always true, the system of
equations has an infinite number of solutions.
8-8 Using Algebra to Solve Linear Systems
Check It Out: Example 2A
Solve each system of equations.
2x + y = 0
2x + 3y = 8
8-8 Using Algebra to Solve Linear Systems
Check It Out: Example 2B
Solve the system of equations.
y = x –1
–3x + 3y = 4