Systems of Linear Equations in Two Variables

1. 1
Systems of Linear Equations
• 4-1 Systems of Linear Equations in Two Variables

2. 2
4-1 Systems of Linear Equations in Two Variables
Deciding whether an ordered pair is a
solution of a linear system.
The solution set of a linear system of equations contains all
ordered pairs that satisfy all the equations at the same time.
• Example 1: Is the ordered pair a solution of the given system?
2x + y = -6 Substitute the ordered pair into each equation.
x + 3y = 2 Both equations must be satisfied.
A) (-4, 2) B) (3, -12)
2(-4) + 2 = -6 2(3) + (-12) = -6
(-4) + 3(2) = 2 (3) + 3(-12) = 2
-6 = -6 -6 = -6
2 = 2 -33 ≠ -6
∴ Yes ∴ No

3. 3
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems by Graphing.
One way to find the solution set of a linear system of equations is to
graph each equation and find the point where the graphs intersect.
• Example 1: Solve the system of equations by graphing.
A) x + y = 5 B) 2x + y = -5
2x - y = 4 -x + 3y = 6
Solution: {(3,2)} Solution: {(-3,1)}

4. 4
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems by Graphing.
There are three possible solutions to a system of linear equations in two
variables that have been graphed:
• 1) The two graphs intersect at a single point. The coordinates give the
solution of the system. In this case, the solution is “consistent” and the
equations are “independent”.
• 2) The graphs are parallel lines. (Slopes are equal) In this case the system
is “inconsistent” and the solution set is 0 or null.
• 3) The graphs are the same line. (Slopes and y-intercepts are the same) In
this case, the equations are “dependent” and the solution set is an infinite
set of ordered pairs.

5. 5
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Elimination.
Remember: If a=b and c=d, then a + c = b + d.
Step 1: Write both equations in standard form
Step 2: Make the coefficients of one pair of variable terms opposite
(Multiply one or both equations by appropriate numbers so that the
sum of the coefficients of either x or y will be zero.)
Step 3: Add the new equations to eliminate a variable
Step 4: Solve the equation formed in step 3
Step 5: Substitute the result of Step 4 into either of the original
equations and solve for the other value.
Step 6: Check the solution and write the solution set.

6. 6
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Elimination.
• Example 2: Solve the system : 2x + 3y = 19
Step 1: Both equations are in standard form 3x - 7y = -6
Step 2: Choose the variable x to eliminate: Multiply the top equation by 3, the bottom
equation by -2
3[2x + 3y = 19] 6x + 9y = 57
-2[3x - 7y = -6] -6x +14y = 12
Step 3: Add the new equations to eliminate a variable
0x + 23y = 69
Step 4: Solve the equation formed in step y = 3
Step 5: Substitute the result of Step 4 into either of the original equations and solve for the
other value. 2x + 3(3) = 19
2x = 10
x = 5 Solution Set: {(5,3)}
Step 6: Check the solution and write the solution set.

7. 7
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Elimination.
• Example 3:
Solve the system :
2[2x - 3y = 1] 4x - 6y = 2
-3[3x - 2y = 9] -9x + 6y = -27
-5x + 0y = -25
x = 5 3(5) - 2y = 9
-2y = -6
Solution Set: {(5,3)} y = 3
1 1 1
rewrite as 6[ ] 2 3 1
3 2 6
: 2 3 1
1 1 1
3 2 6
3 2 9
3 2 9
x y x y
Solv
x y
x y
e x y
x y
⇒ − = ⇒ − =
− =
−
− =
=
=
−

8. 8
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Elimination.
• Example 4:
Solve the system : 2x + y = 6
-8x - 4y = -24
4[2x + y = 6] 8x + 4y = 24
-8x -4y = -24 -8x - 4y = -24
0 = 0 True
Solution Set: {(x,y)| 2x + y = 6}
Note: When a system has dependent equations and an infinite number
of solutions, either equation can be used to produce the solution set.
Answer is given in set-builder notation.

9. 9
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables
by Method of Elimination.
• Example 5:
Solve the system : 4x - 3y = 8
8x - 6y = 14
-2[4x - 3y = 8] -8x + 6y = -16
8x - 6y = 14 8x - 6y = 24
0 = 8 False
Solution Set: 0 or null
Note: There are no ordered pairs that satisfy both
equations. The lines are parallel. There is no solution.

10. 10
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables
by Method of Substitution.
Step 1: Solve one of the equations for either variable
Step 2: Substitute for that variable in the other equation
(The result should be an equation with just one variable)
Step 3: Solve the equation from step 2
Step 4: Substitute the result of Step 3 into either of the
original equations and solve for the other value.
Step 6: Check the solution and write the solution set.

11. 11
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Substitution.
• Example 6: Solve the system : 4x + y = 5
2x - 3y =13
Step 1: Choose the variable y to solve for in the top equation:
y = -4x + 5
Step 2: Substitute this variable into the bottom equation
2x - 3(-4x + 5) = 13 2x + 12x - 15 = 13
Step 3: Solve the equation formed in step 2
14x = 28 x = 2
Step 4: Substitute the result of Step 3 into either of the original equations and solve for the
other value. 4(2) + y = 5
y = -3
Solution Set: {(2,-3)}
Step 5: Check the solution and write the solution set.

12. 12
4-1 Systems of Linear Equations in Two Variables
Solving Linear Systems of two variables by
Method of Substitution.
• Example 7:
Solve the system :
y = -2x + 2
-2x + 5(-2x + 2) = 22 -2x - 10x + 10 = 22
-12x = 12
x = -1 2(-1) + y = 2
y = 4
Solution Set: {(-1,4)}
1 1 1
2 4 2
2
1 1 1
rewrite as 4[ ] 2 2
2 4 2
: 2 2
-2 5 2
5 2
2
2
x y
x y
x y x y
Solve x y
x y
⇒ + = ⇒ + =
+
+
=
+
=
=
=
+
−