Download How to Solve a System of Equations by Algebraic Reasoning

How to Solve a System of Equations by Algebraic Reasoning
(a.k.a. Elimination, Linear Combination, and/or Substitution)
Example 1
Solve by Using Substitution
Use substitution to solve the system of equations.
2x – y = –5
3y – 5x = 14
Solve the first equation for y in terms of x.
2x – y = –5
–y = –5 – 2x
y = 5 + 2x
First equation
Subtract 2x from each side.
Multiply each side by –1.
Substitute 5 + 2x for y in the second equation and solve for x.
3y – 5x = 14
3(5 + 2x) – 5x = 14
15 + 6x – 5x = 14
15 + x = 14
x = –1
Second equation
Substitute 5 + 2x for y.
Distributive Property
Simplify.
Subtract 15 from each side.
Now, substitute the value for x in either original equation and solve for y.
First equation
2x – y = -5
Replace x with –1.
2(–1) – y = -5
Simplify.
–2 – y = -5
Add 2 to each side.
–y = -3
Multiply each side by –1.
y =3
The solution of the system is (–1, 3).
Standardized Test EXAMPLE
Example 2 Solve by Substitution
GRIDDABLE Maryn bought chairs and desks for her employees. The price of the chairs were
$95 each and the price of the desks were $325 each. She bought a total of 23 desks and chairs and
spent $4025. How many chairs did she buy? Show your work.
Read the Test Item
You are asked to find the number of chairs.
Solve the Test Item
Step 1 Define variables and write the system of equations. Let x represent the number of chairs and y
represent the number of desks.
x + y = 23
The total number of desks
and chairs was 23.
The total price was $4025.
95x + 325y = 4025
Step 2
Solve one of the equations for one of the variables in terms of the other. Since the coefficient of
y is 1 and you are asked to find the value of x, it makes sense to solve the first equation for y in terms of x.
x + y = 23
y = 23 – x
Step 3
First equation
Subtract x from each side.
Substitute 23 – x for y in the second equation.
95x + 325y = 4025
95x + 325(23 – x) = 4025
95x + 7475 – 325x = 4025
-230x + 7475 = 4025
-230x = -3450
x = 15
Second equation
Substitute 23 - x for y.
Distributive Property
Simplify.
Subtract 7475 from each
side.
Divide each side by -230.
Step 4 Maryn bought 15 chairs for her employees.
Record your answer in a bubble grid like the one at the
right.
1
5
Example 3
Solve by Using Elimination
Use the elimination method to solve the system of equations.
3p + 4q = 0
p – 4q = –8
The coefficients of q for the first equation and the second equation are additive inverses. This means that
when you add them the result will be 0 and the variable q will be eliminated.
3p + 4q =
(+) p – 4q =
4p
=
p=
0
-8
-8
-2
Add the equations.
Divide each side by 4.
Now find q by substituting –2 for p in either original equation.
3p + 4q = 0
3(–2) + 4q = 0
–6 + 4q = 0
4q = 6
q=
3
First equation
Replace p with –2.
Multiply.
Add 6 to each side.
Divide each side by 4 and simplify.
2
The solution is
2,
3
.
2
Example 4
Multiply, Then Use Elimination
Use the elimination method to solve the system of equations.
2x + 3y = 6
5x – 5y = 65
Multiply the first equation by 5 and the second equation by 2. Then subtract the equations to eliminate
the x variable.
2x + 3y = 6
5x – 5y = 65
Multiply by 5.
Multiply by 2.
10x + 15y = 30
(–) 10x – 10y = 130
25y = –100
y = –4
Replace y with –4 and solve for x.
2x + 3y = 6
2x + 3(–4) = 6
2x – 12 = 6
2x = 18
x=9
First equation
Replace y with –4.
Multiply.
Add 12 to each side.
Divide each side by 2.
The solution is (9, –4).
Subtract the equations.
Divide each side by 25.
Example 5 Inconsistent System
Use the elimination method to solve the system of equations.
4x – y = -7
1
2x - y = 3
2
Use multiplication to eliminate x
4x – y = -7
1
2x - y = 3
2
4x – y = -7
Multiply by 2.
4x – y = 6
0 = -1
Add the equations.
Since there are no values of x and y that will make the equation 0 = -1 true, there
are no solutions for this system of equations.