![Page 1: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/1.jpg)
Solving Linear Equations
![Page 2: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/2.jpg)
To Solve an Equation means...
• To isolate the variable having a coefficient of 1 on one side of the equation.
Examples• x = 5 is solved for x.• y = 2x - 1 is solved for y.
![Page 3: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/3.jpg)
Solving Equations Using Addition and Subtraction
![Page 4: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/4.jpg)
Addition Property of Equality
For any numbers a, b, and c, if a = b, then a + c = b + c.
What it means:
You can add any number to BOTH sides of an equation and the equation will still hold true.
![Page 5: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/5.jpg)
An easy example:
We all know that 7 =7.
Does 7 + 4 = 7? NO!
But 7 + 4 = 7 + 4.
The equation is still
true if we add 4
to both sides.
![Page 6: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/6.jpg)
Let’s try another example!
x - 6 = 10
Add 6 to each side.
x - 6 = 10 +6 +6 x = 16
• Always check your solution!!
• The original problem is x - 6 = 10.
• Using the solution x=16,Does 16 - 6 = 10?
• YES! 10 = 10 and our solution is correct.
![Page 7: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/7.jpg)
What if we see y + (-4) = 9?
Recall that y + (-4) = 9
is the same as y - 4 = 9.
Now we can use the addition property.
y - 4 = 9
+4 +4
y = 13
• Check your solution!
• Does 13 - 4 = 9?• YES! 9=9 and
our solution is correct.
![Page 8: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/8.jpg)
How about -16 + z = 7?• Remember to always
use the sign in front of the number.
• Because 16 is negative, we need to add 16 to both sides.
• -16 + z = 7
+16 +16
z = 23
• Check you solution!
• Does -16 + 23 = 7?
• YES! 7 = 7 and our solution is correct.
![Page 9: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/9.jpg)
A trick question...-n - 10 = 5
+10 +10
-n = 15• Do we want -n? NO,
we want positive n.• If the opposite of n
is positive 15, then n must be negative 15.
• Solution: n = -15
• Check your solution!• Does -(-15)-10=5?• Remember, two
negatives = a positive• 15 - 10 = 5 so our
solution is correct.
![Page 10: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/10.jpg)
Subtraction Property of Equality
• For any numbers a, b, and c, if a = b, then a - c = b - c.
What it means:• You can subtract any number from
BOTH sides of an equation and the equation will still hold true.
![Page 11: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/11.jpg)
3 Examples:1) x + 3 = 17
-3 -3
x = 14• Does 14 + 3 = 17?
2) 13 + y = 20
-13 -13
y = 7• Does 13 + 7 = 20?
3) z - (-5) = -13• Change this
equation. z + 5 = -13
-5 -5
z = -18• Does -18 -(-5) = -13?• -18 + 5 = -13• -13 = -13 YES!
![Page 12: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/12.jpg)
Try these on your own...
x + 4 = -10 x – 14 = -5
y – (-9) = 4 3 – y = 7
12 + z = 15 -5 + z = -7
![Page 13: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/13.jpg)
The answers...
x = -14 x = 9
y = -5 y = -4
z = 3 z = -2
![Page 14: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/14.jpg)
Solving Equations Using Multiplication and Division
![Page 15: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/15.jpg)
An easy example:
We all know that 3 = 3.
Does 3 4 = 3? NO!
But 3 4 = 3 4.
The equation is still true if we multiply both sides by 4.
![Page 16: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/16.jpg)
Let’s try another example!
x = 4 2
Multiply each side by 2.
2 x = 4 2 2x = 8
• Always check your solution!!
• The original problem is
x = 4 2• Using the solution x = 8,
Is x/2 = 4?• YES! 4 = 4 and our
solution is correct.
![Page 17: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/17.jpg)
A fraction times a variable:
The two step method:Ex: 2x = 4 31. Multiply by 3.(3)2x = 4(3) 32x = 12
2. Divide by 2.2x = 12 2 2x = 6
The one step method:
Ex: 2x = 4 31. Multiply by the
RECIPROCAL.
(3)2x = 4(3)(2) 3 (2)
x = 6
![Page 18: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/18.jpg)
x
5
x5
x
5
• The two negatives will cancel each other out.
• The two fives will cancel each other out.
(-5) (-5)
• x = -15• Does -(-15)/5 = 3?
What do we do with negative fractions?
Recall that
Solve .
Multiply both sides by -5.
x5
3
x5
3
![Page 19: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/19.jpg)
Try these on your own...
x = 3 7
4w = 16
y = 8 -2
2x = 12 3
-2z = -12 3x = 9 -4
![Page 20: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/20.jpg)
Division Property of Equality
For any numbers a, b, and c (c ≠ 0), if a = b, then a/c = b/c
What it means: You can divide BOTH sides of an
equation by any number - except zero- and the equation will still hold true.
![Page 21: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/21.jpg)
2 Examples:
1) 4x = 24
Divide both sides by 4.
4x = 24
4 4
x = 6
• Does 4(6) = 24?
YES!
2) -6x = 18
Divide both sides by -6.
-6y = 18
-6 -6
y = -3
• Does -6(-3) = 18? YES!
![Page 22: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/22.jpg)
The answers...
x = 21 w = 4
y = -16 x = 18
z = 6 x = -12
![Page 23: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/23.jpg)
Solving Equations with the Variable on Both Sides
![Page 24: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/24.jpg)
To solve these equations,
•Use the addition or subtraction property to move all variables to one side of the equal sign.
•Solve the equation using the methods we mentioned.
![Page 25: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/25.jpg)
Let’s see a few examples:
1) 6x - 3 = 2x + 13
-2x -2x
4x - 3 = 13
+3 +3
4x = 16
4 4
x = 4
Be sure to check your answer!
6(4) - 3 =? 2(4) + 13
24 - 3 =? 8 + 13
21 = 21
![Page 26: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/26.jpg)
Let’s try another!
2) 3n + 1 = 7n - 5
-3n -3n
1 = 4n - 5
+5 +5
6 = 4n
4 4
Reduce! 3 = n
2
Check:
3(1.5) + 1 =? 7(1.5) - 5
4.5 + 1 =? 10.5 - 5
5.5 = 5.5
![Page 27: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/27.jpg)
Here’s a tricky one!
3) 5 + 2(y + 4) = 5(y - 3) + 10
• Distribute first.
5 + 2y + 8 = 5y - 15 + 10• Next, combine like
terms.
2y + 13 = 5y - 5• Now solve. (Subtract
2y.)
13 = 3y - 5 (Add 5.)
18 = 3y (Divide by 3.)
6 = y
Check:
5 + 2(6 + 4) =? 5(6 - 3) + 10
5 + 2(10) =? 5(3) + 10
5 + 20 =? 15 + 10
25 = 25
![Page 28: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/28.jpg)
Let’s try one with fractions!
4)
3 - 2x = 4x - 6
3 = 6x - 6
9 = 6x so x = 3/2
3
8
1
4x
1
2x
3
4Steps:• Multiply each termby the least common denominator (8) to eliminate fractions.
• Solve for x.• Add 2x.• Add 6.• Divide by 6.
(8)3
8 (8)
1
4x(8)
1
2x (8)
3
4
![Page 29: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/29.jpg)
Two special cases:
6(4 + y) - 3 = 4(y - 3) + 2y
24 + 6y - 3 = 4y - 12 + 2y
21 + 6y = 6y - 12
- 6y - 6y
21 = -12 Never true!
21 ≠ -12 NO SOLUTION!
3(a + 1) - 5 = 3a - 2
3a + 3 - 5 = 3a - 2
3a - 2 = 3a - 2
-3a -3a
-2 = -2 Always true!
We write IDENTITY.
![Page 30: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/30.jpg)
Try a few on your own:
• 9x + 7 = 3x - 5
• 8 - 2(y + 1) = -3y + 1
• 8 - 1 z = 1 z - 7
2 4
![Page 31: Solving Linear Equations To Solve an Equation means... To isolate the variable having a coefficient of 1 on one side of the equation. Examples x = 5](https://reader036.vdocuments.us/reader036/viewer/2022070412/56649ef05503460f94c00351/html5/thumbnails/31.jpg)
The answers:
• x = -2
• y = -5
• z = 20