Solving Linear Equations

1. Math 80
Linear Equations in One
Variable

2. Solving Linear Equations

3. Solve.
1) 5x + 4 = 39 3) 7 - x = 12
2) 4)
x
3
4 2  13 6
5
 
x
-4 -4
5x = 35
5 5
x = 7
+4 +4
x
3
6(3) (3)
x = 18
-7 -7
- x = 5
x = -5
-6 -6
7
5
 
x
(-5) (-5)
-35 = x
-1 -1

4. Solving equations multi-step
equations

5. Objective - To solve multi-step equations.
Combine like terms Distribute
4x x 8 30  
3x 8 30 
8 8 
3x 22
3 3
22x
3
 17
3

5(y 4) 12 
5y 20 12 
20 20 
5y 32
5 5
32y
5
 26
5


6. (-1)
Solve.
1) 3(x - 2) = 17 3) 12 - 2(x + 4) = 28
2) -(5 - x) = 9 4)
5
4
10


y
3x - 6 = 17
+6 +6
3x = 23
3 3
x 
23
3
 7
2
3
-5 + x = 9
+5 +5
x = 14
12 - 2x - 8 = 28
-2x + 4 = 28
-4 -4
-2x = 24
-2 -2
x = -12
(4) (4)
5 - y = 40
-5 -5
-y = 35(-1)
y = -35

7. Solve.
6(x + 4) - 2(x - 7) = 10
6x + 24 - 2x + 14 = 10
4x + 38 = 10
-38 -38
4x = -28
4 4
x = -7

8. Solve.
8 - 4[x - (2x + 4)] = 16
8 - 4[x - 2x - 4] = 16
8 - 4[-x - 4] = 16
8 + 4x + 16 = 16
4x + 24 = 16
-24 -24
4x = -8
4 4
x = -2

9. Linear equations with variables
on both sides

10. -4 - 4
Solve. 2x + 4 = 6x - 20
-6x -6x
-4x + 4 = -20
-4 -4
-4x = -24
x = 6

11. 9 9
Solve. 3x - 5 = -6x + 1
+6x +6x
9x - 5 = 1
+5 +5
9x = 6
2
3
x 

12. -1 -1
Solve. -4x - 8 = -3x + 7
+3x +3x
-x - 8 = 7
+8 +8
-x = 15
15x  

13. Solve.
1) 3(x - 2) - 4x = 5(x - 3)
3x - 6 - 4x = 5x - 15
-x - 6 = 5x - 15
+x +x
-6 = 6x - 15
+15 +15
9 = 6x
6 6
x 
9
6

3
2

14. Solve.
2) 2(3 + k) - 3(8k + 4) = 2(k - 3) - (8k - 5)
6 + 2k - 24k - 12 = 2k - 6 - 8k + 5
-22k - 6 = -6k - 1
+6k +6 +6k +6
-16k = 5
k  
5
16

15. 3) 4(m - 2) = 8 - 4[3(m - 5) - 5m]
4m - 8 = 8 - 4[3m - 15 - 5m]
4m - 8 = 8 - 4[-2m - 15]
4m - 8 = 8 + 8m + 60
4m - 8 = 8m + 68
-4m -4m
-8 = 4m + 68
-68 -68
-76 = 4m
4 4 m  -19

16. Linear equations with one
fraction

17. Solving Equations Involving Fractions
2
9 11
5
x  
9  9
2
2
5
x 
5 2 5
2
2 5 2
x
   
   
   
5x 
Check !

18. Solve.
1) 2)
3) 4)
10
3
4
28 x
-10 -10
 
3
4
18x 4
3
  4
3
 3 x
4

x = -24
5
6
4 11x  
+4 +4
5
6
15x  6
5  6
5
x = 18

 
2
3
9 21x
-9 -9


2
3
12x 3
2

 3
2

x = -18
29
3
4
5 x
-5 -5
24
3
4
 x 4
3  4
3
32 = x

19. Solve.
1)
2)
3)
4)
2
5 12
7
y  
7
2
 
 
 
7
2
 
 
 
2
7
7
y 
49
2
y 
2
9 18
3
m

  
3
2
 
 
 
3
2
 
 
 
2
9
3
m

 
27
2
m 
2
7 9
7
x   
4
9 3
5
m

  
7 x 
15m  
2
2
7
x 
7
2
 
 
 
7
2
 
 
 
4
12
5
m  
5
4
 
 
 
5
4
 
 
 

20. Linear equations with more
than one fraction

21. Solve:
2 1 3x
3 2 8
  
LCD 24
2 1 3x
3 2 8
24   
  
16x 12 9  
12 12 
21x
16
 
16x 21 
16 16

22. Solve:
2 2 2x
5 15 3
  
LCD 15
2 2 2x
5 15
5
3
1    
  
6x 2 10  
2 2 
4x
3
 
6x 8 
6 6 

23. Solve:
1 1 1 5x x
2 4 3 4
  
LCD 12
1 1 1 5x x
2 4 3
12
4
   
  
6x 3 4x 15  
4x 4x 
2x 3 15 
3 3 
2x 12
2x 12
2 2
x 6

24. Solve:
3 5(5x 10) (12x 18)
5 6
  
LCD 30
3 5(5x 10) (12x30 18)
5 6
   
  
18(5x 10) 25(12x 18)  
90x 90x 
180 210x 450 
450 +450
630 210x
630 210x
210 210
3 x
90x 180 300x 450  

25. Linear equations with Decimals

26. Solve:
0.35x 0.2 0.15x 0.1  
Multiply by 100 to eliminate the decimal
 0.35x 0.2 0.15x .0 010 1  
35x 20 15x 10  
15x 15x 
3x
2

20x 20 10 
20 20
20 +20
20x 30

27. Solve:
0.25x 0.05 0.02x 0.15  
Multiply by 100 to eliminate the decimal
 0.25x 0.05 0.2x1 0 .0 0 15  
25x 5 20x 15  
20x 20x 
x 4
5x 5 15 
5 5
5 +5
5x 20

28. Linear equations with no
solution

29. False Statement
Solve.
6x - 8 = 6x + 3
-6x -6x
- 8 = 3
No Solution

30. Solve.
-15 + 3x = 3(x - 5)
-15 + 3x = 3x - 15
-3x -3x
-15 = - 15
True Statement
Infinitely Many Solutions
Or all Real Numbers
Or All Reals

31. Find the dimensions of the rectangle below if the
area is 125 m2 .
5x - 15
5 Area •l w
125 (5x  15)5
125  25x  75
75  75
200  25x
8  x
= 255(8) - 15
5, 25