Pre Algebra
Section 4.2
Unit 4 - Equations
Section 2 – Solving One-Step Equations
In this section we will be looking for solutions to equations.
A solution is a number that can be plugged into an equation that keeps the equation true.
Is ‫ = ݔ‬3 a solution to ‫ ݔ‬+ 7 = 10?
‫ ݔ‬+ 7 = 10
3 + 7 = 10
10 = 10
Since this is a true statement , ࢞ = ૜ is a solution to ࢞ + ૠ = ૚૙.
Is ‫ = ݔ‬2 a solution to ‫ ݔ‬+ 7 = 10?
‫ ݔ‬+ 7 = 10
2 + 7 = 10
9 = 10
Since 9 ≠ 10, ࢞ = ૛ is NOT a solution of ࢞ + ૠ = ૚૙.
We can solve equations and find solutions as long as we keep the equations balanced as we work the
problem. Think of the “=” a the tip of a balance or a scale. Adding pebbles to one side of a scale would
through it off balance. In order to not change the balance you have to add pebbles to the other side of
the scale as well.
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Pre Algebra
Section 4.2
When I solve equations the first thing I do is look for my variable and think about how the numbers are
attached. I then do the opposite to “unattach” them.
Solving Equations of the Form ࢞ + ࢇ = ࢈
For problems that have a number added to the variable – I do the opposite of add – I subtract the
number from both sides.
Example 1)
Solve ࢞ + ૝ = ૚૞
‫ ݔ‬+ 4 = 15
−4 − 4
‫ ݔ‬+ 0 = 11
‫ = ݔ‬11
We can Subtract 4 to get x by itself, but we
must do it to both sides of the = sign.
The Solution is 11 .
We can check to be sure this is correct by plugging ‫ = ݔ‬11 into the original problem.
‫ ݔ‬+ 4 = 15
11 + 4 = 15
15 = 15
Since the result is the same on both sides the solution is correct.
For problems that have a number subtracted from the variable – I do the opposite of subtract – I add the
number to both sides.
Example 2)
Solve ࢟ − ૠ = ૚૛
‫ ݕ‬− 7 = 12
+7 + 7
‫ ݕ‬+ 0 = 19
We can add 7 to get y by itself, but we
must do it to both sides of the = sign.
‫ = ݕ‬19
Check: This solution is correct since 19 – 7 = 12.
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Pre Algebra
Section 4.2
Example 3)
࢈ − ૜ = −૚૞
ܾ − 3 = −15
+3
+3
ܾ + 0 = −12
We can add 3 to get b by itself, but we
must do it to both sides of the = sign.
ܾ = −12
Check: This solution is correct since −12 − 3 = −15.
Solving Equations of the Form ࢇ࢞ = ࢈
For problems that have a number multiplied to the variable – I do the opposite of multiply – I divide by
the number on both sides.
Example 4)
૞࢞ = ૛૞
5‫ = ݔ‬25
5
5
1‫ = ݔ‬5
We can divide by 5 to get x by itself, but
we must do it to both sides of the = sign.
‫=ݔ‬5
Check: The solution is correct since 5 ∙ 5 = 25.
Example 5 )
−ૠ࢟ = ૝૛
−7‫ = ݕ‬42
−7 − 7
We can divide by -7 to get y by itself, but
we must do it to both sides of the = sign.
1‫ = ݕ‬−6
‫ = ݕ‬−6
Check: The solution is correct since −7 ∙ −6 = 42.
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Pre Algebra
Section 4.2
Example 6)
−૜ࢇ = −૜૟
−3ܽ = −36
−3
−3
We can divide by 5 to get a by itself, but
we must do it to both sides of the = sign.
ܽ = 12
Check: The solution is correct since −3 ∙ 12 = −36.
What do you do to get x by itself?
Example 7) Consider the following problems
a) ૛࢞ = −૚૙
b)
࢞ − ૚૙ = −૚૞
a) ૛࢞ = −૚૙
Since the 2 is held to the x by multiply – I will do the opposite of multiply and divide.
2‫ = ݔ‬−10
2‫ = ݔ‬−10
2
2
1‫ = ݔ‬−5
Check:
2 ∙ −5 = 10
x = −5
࢈)
࢞ − ૚૙ = −૚૞
Since the 10 is being subtracted from the x – I will do the opposite of subtract and add.
x − 10 = −15
x − 10 = −15
+10 + 10
Check:
−5 − 10 = −15
x + 0 = −5
x = −5
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Pre Algebra
Section 4.2
If there are like terms on the same side of the “=”, we begin by combining them before considering what
operation to do to both sides.
Example 8)
࢞ + ૡ = ૚૛ − ૞
‫ ݔ‬+ 8 = 12 − 5
We combine like terms before getting x by itself.
‫ݔ‬+8 = 7
−8 − 8
‫ ݔ‬+ 0 = −1
‫ = ݔ‬−1
Check: The solution is correct since −1 + 8 = 12 − 5.
Example 9)
ૠ࢞ − ૜࢞ = ૠ૙ − ૝૟
7‫ ݔ‬− 3‫ = ݔ‬70 − 46
First combine like terms.
4‫ = ݔ‬24
4
4
‫=ݔ‬6
Check: The solution is correct since 7 ∙ 6 − 3 ∙ 6 = 70 − 46.
Words to Equations
It is essential to understand how words can build equations. Consider the following:
A number decreased by 7 is 15.
݊ − 7 = 15
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Pre Algebra
Section 4.2
Words that mean math – add to the list as you come across new words!
+
Add to
Increased by
Decreased by
X
÷
Product
of
Quotient
Divided Among
Twice (X 2)
Half (÷2)
=
is
Example10)
The product of a number and three is twenty – seven, find the number.
Equation:
3 ∙ ݊ = 27
Product means multiply 3 and the number “n”.
3݊ = 27
3
3
݊=9
The number is 9.
Check: 3 ∙ 9 = 27
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Pre Algebra
Section 4.2
Exercise 4.2
NAME:___________________________________
Solve the following.
1. ‫ ݔ‬+ 4 = 7
2. ‫ ݔ‬+ 7 = 17
3. ‫ ݔ‬+ 8 = 22
Check:
Check:
Check:
4. ‫ ݔ‬− 13 = 10
5. ‫ ݔ‬− 4 = 15
6. ‫ ݔ‬− 3 = 7
Check:
Check:
Check:
7. ‫ ݔ‬+ 10 = −15
8. ‫ ݔ‬− 3 = −10
9. ‫ ݔ‬− 1 = −12
Check:
Check:
Check:
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Section 4.2
10. 2‫ = ݔ‬12
11. 3‫ = ݔ‬24
12. 5‫ = ݔ‬45
Check:
Check:
Check:
13. −4‫ = ݔ‬16
14. −7‫ = ݔ‬42
15. −3‫ = ݔ‬−15
Check:
Check:
Check:
16.−4‫ = ݔ‬−24
17. −9‫ = ݔ‬−72
18. −8‫ = ݔ‬32
Check:
Check:
Check:
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Pre Algebra
Section 4.2
Solve
19. ‫ ݔ‬− 3 = −13
20. −4‫ = ݔ‬−16
21. −2‫ = ݔ‬−18
22. 5‫ ݔ‬− 3‫ = ݔ‬10
23. −4‫ ݔ‬− 7‫ = ݔ‬−33
24. −5‫ ݔ‬+ 8‫ = ݔ‬27
25. 5‫ = ݔ‬−72 + 47
26. 9‫ = ݔ‬−42 − 3
27. −3‫ = ݔ‬25 − 16
28. ‫ ݔ‬− 10‫ ݔ‬+ 12‫ = ݔ‬13 − 7
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29.
173
4‫ ݔ‬+ 3‫ ݔ‬− 5‫ = ݔ‬54 − 12
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Pre Algebra
Section 4.2
30. 15‫ ݔ‬− 7‫ = ݔ‬−12 − 52
Write an equation for the following and then solve.
31. Twice a number is 16, find the number.
Equation:
32. A number increased by 20 is 35, find the number.
Equation:
33. The product of a number and 5 is 55, find the number.
Equation:
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