11.3: Solving Equations of the Form x + a = b and x - a = b
- understand the meaning and function of an equation
- understand what is meant by the solution to an equation
- be able to solve equations of the form \(x + a = b\) and \(x - a = b\)
Equations
Equation
An equation is a statement that two algebraic expressions are equal.
The following are examples of equations:
\(\begin{array} {c} {\underbrace{x + 6}_{\text{This}}} & = & {\underbrace{10}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}\) \(\begin{array} {c} {\underbrace{x - 4}_{\text{This}}} & = & {\underbrace{-11}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}\) \(\begin{array} {c} {\underbrace{3y - 5}_{\text{This}}} & = & {\underbrace{-2 + 2y}_{\text{This}}} \\ {^\text{expression}} & ^= & {^\text{expression}} \end{array}\)
Notice that \(x + 6\), \(x - 4\), and \(3y - 5\) are not equations. They are expressions. They are not equations because there is no statement that each of these expressions is equal to another expression.
Solutions and Equivalent Equations
Conditional
Equations
The truth of some equations is conditional upon the value chosen for the variable. Such equations are called
conditional equations
. There are two additional types of equations. They are examined in courses in algebra, so we will not consider them now.
Solutions and Solving an
Equation
The set of values that, when substituted for the variables, make the equation true, are called the
solutions
of the equation.
An equation has been
solved
when all its solutions have been found.
Verify that 3 is a solution to \(x + 7 = 10\).
Solution
When \(x = 3\),
becomes \(\begin{array} {rcll} {x + 7} & = & {10} & {} \\ {3 + 7} & = & {10} & {} \\ {10} & = & {10} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{3 is a solution to } x + 7 = 10} \end{array}\)
Verify that -6 is a solution to \(5y + 8 = -22\).
Solution
When \(y = -6\),
becomes \(\begin{array} {rcll} {5y + 8} & = & {-22} & {} \\ {5(-6) + 8} & = & {-22} & {} \\ {-30 + 8} & = & {-22} & {} \\ {-22} & = & {-22} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{-6 is a solution to } 5y + 8 = -22} \end{array}\)
Verify that 5 is not a solution to \(a - 1 = 2a + 3\).
Solution
When \(a = 5\),
becomes \(\begin{array} {rcll} {a - 1} & = & {2a + 3} & {} \\ {5 - 1} & = & {2 \cdot 5 + 3} & {} \\ {5 - 1} & = & {10 + 3} & {} \\ {4} & = & {13} & {\text{a } false \text{ statement, verifying that 5}} \\ {} & & {} & {\text{is not a solution to } a - 1 = 2a + 3} \end{array}\)
Verify that -2 is a solution to \(3m - 2 = -4m - 16\).
Solution
When \(m = -2\),
becomes \(\begin{array} {rcll} {3m - 2} & = & {-4m - 16} & {} \\ {3(-2) - 2} & = & {-4(-2) - 16} & {} \\ {-6 - 2} & = & {8 - 16} & {} \\ {-8} & = & {-8} & {\text{which is a } true \text{ statement, verifying that}} \\ {} & & {} & {\text{-2 is a solution to } 3m - 2 = -4m - 16} \end{array}\)
Practice Set A
Verify that 5 is a solution to \(m + 6 = 11\).
- Answer
-
Substitute 5 into \(m + 6 = 11\).
Thus, 5 is a solution.
Practice Set A
Verify that −5 is a solution to \(2m - 4 = -14\).
- Answer
-
Substitute -5 into \(2m - 4 = -14\).
Thus, -5 is a solution.
Practice Set A
Verify that 0 is a solution to \(5x + 1 = 1\).
- Answer
-
Substitute 0 into \(5x + 1 = 1\).
Thus, 0 is a solution.
Practice Set A
Verify that 3 is not a solution to \(-3y + 1 = 4y + 5\).
- Answer
-
Substitute 3 into \(-3y + 1 = 4y + 5\).
Thus, 3 is not a solution.
Practice Set A
Verify that -1 is a solution to \(6m - 5 + 2m = 7m - 6\).
- Answer
-
Substitute -1 into \(6m - 5 + 2m = 7m - 6\).
Thus, -1 is a solution.
Equivalent Equations
Some equations have precisely the same collection of solutions. Such equations are called equivalent equations. For example, \(x - 5 = -1, x + 7 = 11,\) and \(x = 4\) are all equivalent equations since the only solution to each is \(x = 4\). (Can you verify this?)
Solving Equations
We know that the equal sign of an equation indicates that the number represented by the expression on the left side is the same as the number represented by the expression on the right side.
| This number | is the same as | this number |
| \(\downarrow\) | \(\downarrow\) | \(\downarrow\) |
| \(x\) | = | 4 |
| \(x + 7\) | = | 11 |
| \(x - 5\) | = | -1 |
Addition/Subtraction Property of
Equality
From this, we can suggest the
addition/subtraction property of equality
.
Given any equation,
- We can obtain an equivalent equation by adding the same number to both sides of the equation.
- We can obtain an equivalent equation by subtracting the same number from both sides of the equation.
The Idea Behind Equation Solving
The idea behind
equation solving
is to isolate the variable on one side of the equation. Signs of operation (+, -, \(\cdot\), \(\div\)) are used to associate two numbers. For example, in the expression \(5 + 3\), the numbers 5 and 3 are associated by addition. An association can be
undone
by performing the opposite operation. The addition/subtraction property of equality can be used to undo an association that is made by addition or subtraction.
Subtraction is used to undo an addition.
Addition is used to undo a subtraction.
The procedure is illustrated in the problems of [link] .
Use the addition/subtraction property of equality to solve each equation.
\(x + 4 = 6\).
Solution
4 is associated with \(x\) by addition. Undo the association by subtracting 4 from both sides.
\(x + 4 - 4 = 6 - 4\)
\(x + 0 =2\)
\(x = 2\)
Check: When \(x = 2\), \(x + 4\) becomes
The solution to \(x + 4 = 6\) is \(x = 2\).
\(m - 8 = 5\).
Solution
8 is associated with \(m\) by subtraction. Undo the association by adding 8 to both sides.
\(m - 8 + 8 = 5 + 8\)
\(m + 0 = 13\)
\(m = 13\)
Check: When \(m = 13\),
becomes
a true statement.
The solution to \(m - 8 = 5\) is \(m = 13\).
\(-3 - 5 = y - 2 + 8\).
Solution
Before we use the addition/subtraction property, we should simplify as much as possible.
\(-3 - 5 = y - 2 + 8\).
\(-8 = y + 6\)
6 is associated with \(y\) by addition. Undo the association by subtracting 6 from both sides.
\(-8 - 6 = y + 6 - 6\)
\(-14 = y + 0\)
\(-14 = y\)
This is equivalent to \(y = -14\).
Check: When \(y = -14\),
\(-3 - 5 = y - 2 + 8\)
becomes
a true statement.
The solution to \(-3 - 5 = y - 2 + 8\) is \(y = -14\).
\(-5a + 1 + 6a = -2\).
Solution
Begin by simplifying the left side of the equation.
\(\underbrace{-5a + 1 + 6a}_{-5 + 6 = 1} = -2\)
\(a + 1 = -2\) 1 is associated with aa by addition. Undo the association by subtracting 1 from both sides.
\(a + 1 - 1 = -2 - 1\)
\(a + 0 = -3\)
\(a = -3\)
Check: When \(a = -3\),
\(-5a + 1 + 6a = -2\)
becomes
a true statement.
The solution to \(-5a + 1 + 6a = -2\) is \(a = -3\).
\(7k - 4 = 6k + 1\).
Solution
In this equation, the variable appears on both sides. We need to isolate it on one side. Although we can choose either side, it will be more convenient to choose the side with the larger coefficient. Since 8 is greater than 6, we’ll isolate \(k\) on the left side.
\(7 k - 4 = 6k + 1\) Since \(6k\) represents \(+6k\), subtract \(6k\) from each side.
\(\underbrace{7 k - 4 - 6k}_{7 - 6 = 1} = \underbrace{6k + 1 - 6k}_{6 - 6 = 0}\)
\(k - 4 = 1\) 4 is associated with \(k\) by subtraction. Undo the association by adding 4 to both sides.
\(k - 4 + 4 = 1 + 4\0
\(k = 5\)
Check: When \(k = 5\).
\(7k - 4 = 6k + 1\)
becomes
a true statement.
The solution to \(7k - 4 = 6k + 1\) is \(k = 5\)
\(-8 + x = 5\).
Solution
-8 is associated with \(x\) by addition. Undo the by subtracting -8 from both sides. Subtracting -8 we get \(-(-8) = +8\). We actually add 8 to both sides.
\(-8 + x + 8 = 5 + 8\)
\(x = 13\)
Check: When \(x = 13\)
\(-8 + x = 5\)
becomes
a true statement.
The solution to \(-8 + x = 5\) is \(x = 13\).
Practice Set B
\(y + 9 = 4\)
- Answer
-
\(y = -5\)
Practice Set B
\(a - 4 = 11\)
- Answer
-
\(a = 15\)
Practice Set B
\(-1 + 7 = x + 3\)
- Answer
-
\(x = 3\)
Practice Set B
\(8m + 4 - 7m = (-2) (-3)\)
- Answer
-
\(m = 2\)
Practice Set B
\(12k - 4 = 9k - 6 + 2k\)
- Answer
-
\(k = -2\)
Practice Set B
\(-3 + a = -4\)
- Answer
-
\(a = -1\)
Exercises
For the following 10 problems, verify that each given value is a solution to the given equation.
Exercise \(\PageIndex{1}\)
\(x - 11 = 5\), \(x = 16\)
- Answer
-
Substitue \(x = 4\) into the equation \(4x - 11 = 5\).
\(16 - 11 = 5\)
\(5 = 5\)
\(x = 4\) is a solution
Exercise \(\PageIndex{2}\)
\(y - 4 = -6\), \(y = -2\)
Exercise \(\PageIndex{3}\)
\(2m - 1 = 1\), \(m = 1\)
- Answer
-
Substitue \(m = 1\) into the equation \(2m - 1 = 1\).
\(m = 1\) is a solution.
Exercise \(\PageIndex{4}\)
\(5y + 6 = -14\), \(y = -4\)
Exercise \(\PageIndex{5}\)
\(3x + 2 - 7x = -5x - 6\), \(x = -8\)
- Answer
-
Substitue \(x = -8\) into the equation \(3x + 2 - 7 = -5x - 6\).
\(x = -8\) is a solution.
Exercise \(\PageIndex{6}\)
\(-6a + 3 + 3a = 4a + 7 - 3a\), \(a = -1\)
Exercise \(\PageIndex{7}\)
\(-8 + x = -8\), \(x = 0\)
- Answer
-
Substitue \(x = 0\) into the equation \(-8 + x = -8\).
\(x = 0\) is a solution.
Exercise \(\PageIndex{8}\)
\(8b + 6 = 6 - 5b\), \(b = 0\)
Exercise \(\PageIndex{9}\)
\(4x - 5 = 6x - 20\), \(x = \dfrac{15}{2}\)
- Answer
-
Substitue \(x = \dfrac{15}{2}\) into the equation \(4x - 5 = 6x - 20\).
\(x = \dfrac{15}{2}\) is a solution
Exercise \(\PageIndex{10}\)
\(-3y + 7 = 2y - 15\), \(y = \dfrac{22}{5}\)
Solve each equation. Be sure to check each result.
Exercise \(\PageIndex{11}\)
\(y - 6 = 5\)
- Answer
-
\(y = 11\)
Exercise \(\PageIndex{12}\)
\(m + 8 = 4\)
Exercise \(\PageIndex{13}\)
\(k - 1 = 4\)
- Answer
-
\(k = 5\)
Exercise \(\PageIndex{14}\)
\(h - 9 = 1\)
Exercise \(\PageIndex{15}\)
\(a + 5 = -4\)
- Answer
-
\(a = -9\)
Exercise \(\PageIndex{16}\)
\(b - 7 = -1\)
Exercise \(\PageIndex{17}\)
\(x + 4 - 9 = 6\)
- Answer
-
\(x = 11\)
Exercise \(\PageIndex{18}\)
\(y - 8 + 10 = 2\)
Exercise \(\PageIndex{19}\)
\(z + 6 = 6\)
- Answer
-
\(z = 0\)
Exercise \(\PageIndex{20}\)
\(w - 4 = -4\)
Exercise \(\PageIndex{21}\)
\(x + 7 - 9 = 6\)
- Answer
-
\(x = 8\)
Exercise \(\PageIndex{22}\)
\(y - 2 + 5 = 4\)
Exercise \(\PageIndex{23}\)
\(m + 3 - 8 = -6 + 2\)
- Answer
-
\(m = 1\)
Exercise \(\PageIndex{24}\)
\(z + 10 - 8 = -8 + 10\)
Exercise \(\PageIndex{25}\)
\(2 + 9 = k - 8\)
- Answer
-
\(k = 19\)
Exercise \(\PageIndex{26}\)
\(-5 + 3 = h - 4\)
Exercise \(\PageIndex{27}\)
\(3m - 4 = 2m + 6\)
- Answer
-
\(m = 10\)
Exercise \(\PageIndex{28}\)
\(5a + 6 = 4a - 8\)
Exercise \(\PageIndex{29}\)
\(8b + 6 + 2b = 3b - 7 + 6b - 8\)
- Answer
-
\(b = -21\)
Exercise \(\PageIndex{30}\)
\(12h - 1 - 3 - 5h = 2h + 5h + 3(-4)\)
Exercise \(\PageIndex{31}\)
\(-4a + 5 - 2a = -3a - 11 - 2a\)
- Answer
-
\(a = 16\)
Exercise \(\PageIndex{32}\)
\(-9n - 2 - 6 + 5n = 3n - (2) (-5) - 6n\)
Calculator Exercises
Exercise \(\PageIndex{33}\)
\(y - 2.161 = 6.063\)
- Answer
-
\(y = 7.224\)
Exercise \(\PageIndex{34}\)
\(a - 44.0014 = -21.1625\)
Exercise \(\PageIndex{35}\)
\(-0.362 - 0.416 = 5.63m - 4.63m\)
- Answer
-
\(m = -0.778\)
Exercise \(\PageIndex{36}\)
\(8.078 - 9.112 = 2.106y - 1.106y\)
Exercise \(\PageIndex{37}\)
\(4.23k + 3.18 = 3.23k - 5.83\)
- Answer
-
\(k = -9.01\)
Exercise \(\PageIndex{38}\)
\(6.1185x - 4.0031 = 5.1185x - 0.0058\)
Exercise \(\PageIndex{39}\)
\(21.63y + 12.40 - 5.09y = 6.11y - 15.66 + 9.43y\)
- Answer
-
\(y = -28.06\)
Exercise \(\PageIndex{40}\)
\(0.029a - 0.013 - 0.034 -0.057 = -0.038 + 0.56 + 1.01a\)
Exercises for Review
Exercise \(\PageIndex{41}\)
Is \(\dfrac{\text{7 calculators}}{\text{12 students}}\) an example of a ratio or a rate?
- Answer
-
rate
Exercise \(\PageIndex{42}\)
Convert \(\dfrac{3}{8}\)% to a decimal
Exercise \(\PageIndex{43}\)
0.4% of what number is 0.014?
- Answer
-
3.5
Exercise \(\PageIndex{44}\)
Use the clustering method to estimate the sum: \(89 + 93 + 206 + 198 + 91\)
Exercise \(\PageIndex{45}\)
Combine like terms: \(4x + 8y + 12y + 9x - 2y\).
- Answer
-
\(13x + 18y\)