Solving Systems Using
Substitution
Objective: To solve systems
algebraically
Substitution Property of Equality
• If a = b, then a can be used for b in any
situation.
• Ex: 1 = 2
2
So, if we’re adding 1 + 1
2
Then 2 + 1 = 3
we substituted the 2 for the 1
2 2 2
2
Solving for a Variable
• Solving for x:
• 1st … Eliminate the
constant
• Divide by the
coefficient
• Your answer is …
2x – 4 = 12
+4
+4
2x = 16
2
2
x=8
Solving for a Variable (Part II)
• Solving for x:
• 1st … Eliminate the
constant
• Divide by the
coefficient
• Your answer is …
ax – b = c
+b +b
ax = c + b
a
a
x=c+b
a
Systems of Equations
• Two or more equations using the same two or
more variables.
Ex:
2x – y = 8
y=x–3
Notice that both equations have an x and a y
Solving a System of Equations
• Step 1: Find the variable that is already solved.
• Step 2: If no variable is solved for, find out which one
can be solve for easier.
• Step 3: Insert the formula of the solved variable for
the variable in the 2nd equation
• Step 4: Solve for the 2nd Variable
• Step 5: Use the answer to solve for the 1st variable
• Step 6: Answer must come in an ordered pair (x, y)
Example #1
Solve:
2x – y = 8
y=x–3
2x – (x – 3) = 8
2x – x + 3 = 8
x+3=8
-3 -3
x=5
• Step 1: Find the variable
that is already solved
• In this case, y is already
solved for. y = x – 3. So use
x – 3 for y in the top
equation (Step 3)
• Steps 4: Now solve for x
Example #1 (Continued)
x=5
y=x–3
y=5–3
y=2
(5,2)
• Now plug in 5 for x in
either equation. (Step 5)
• Now set answer as an
ordered pair (Step 6).
Example #2
3x + y = 13
4x + 2y = 30
3x + y = 13
-3x
-3x
y = -3x + 13
4x + 2(-3x +13) = 30
4x – 6x + 26 = 30
-2x + 26 = 30
-26 -26
-2x = 4
-2 -2
x = -2
• Step #1: Find the variable that is
already solve.
• Not in this case. Go to Step 2.
• To solve for x in equation #1, you
need to subtract y and divide by 3.
• To solve for y in equation #1, you
need to subtract 3x and that’s it.
• Once you find y, plug it in on the
other equation.
• Now solve for x
• Now that we know x, plug it in
either equation to find y.
Example #2 (Continued)
y = -3x + 13
y = -3(-2) + 13
y = 6 + 13
y = 19
(-2, 19)
• We’ll use this
equation, because it
already gives us an
answer for y.
• Our solution