The Addition and Subtraction Properties of Equality: Videos & Practice Problems

A linear equation, expressed as $ax+b=c$ where $a\ne 0$, represents a first-degree polynomial with terms involving variables and coefficients. Solving such equations involves isolating the variable using properties of equality, like the addition and subtraction properties, which maintain equivalent equations by performing operations on both sides. Solutions are values that satisfy the equation, verified by substitution. Understanding these concepts is foundational for manipulating polynomials, exponents, and expressions in standard form, essential for algebraic proficiency.

Introduction to Linear Equations

A linear equation is a mathematical statement that asserts the equality of two algebraic expressions and can be written in the standard form \(ax + b = c\), where \(a\), \(b\), and \(c\) are real numbers and \(a \neq 0\). The variable, often represented as \(x\) but not limited to it, appears to the first power, which is why linear equations are also called first-degree equations. This means the variable has an implicit exponent of one, distinguishing linear equations from higher-degree polynomials.

The solution to a linear equation is the specific value of the variable that makes the equation true when substituted back into it. For example, in the equation \$2x + 6 = 0\(, substituting \)x = -3\( yields \)2(-3) + 6 = -6 + 6 = 0\(, confirming that \)x = -3\( is a valid solution. Conversely, if substituting a value does not satisfy the equation, such as \)w = -1\( in \)5 = 8w - 3\(, which results in \)5 \neq -11\(, then that value is not a solution.

Linear equations can involve any variable symbol, and even if the equation is not initially presented in the form \)ax + b = c\(, it can often be rearranged to fit this format. For instance, \)5 = 8w - 3\( can be rewritten as \)8w - 3 = 5\(, clearly showing \)a = 8\(, \)b = -3\(, and \)c = 5\(.

The set of all solutions to a linear equation is called the solution set, typically denoted using curly brackets. For example, if \)x = -3\( is the only solution to \)2x + 6 = 0\(, the solution set is written as \)\{ -3 \}$. Understanding how to identify and verify solutions is fundamental before progressing to methods for solving linear equations.

Introduction to Linear Equations Example 1

To determine whether an expression is a linear equation in one variable, it must first be an equation, meaning it has an equal sign separating two expressions. For example, an expression like 4x - 7 + 3 is not an equation because it lacks an equal sign, so it cannot be classified as a linear equation in one variable.

When examining an equation such as 5y - y = 2, it qualifies as an equation since it has two expressions separated by an equal sign. Although the variable y appears twice, both instances are to the first power, which is a key characteristic of linear equations. Combining like terms, 5y - y simplifies to 4y, resulting in the equation 4y = 2. This clearly represents a linear equation in one variable because it involves a single variable raised only to the first power.

In contrast, an equation like 6x + 1 = 2t² is not a linear equation in one variable. This is because it contains two different variables, x and t, and the variable t is squared, which violates the linearity condition. Linear equations must have only one variable, and that variable must be to the first power.

In summary, a linear equation in one variable must have exactly one variable raised to the first power and must be an equation with an equal sign. Expressions without an equal sign or those involving multiple variables or variables raised to powers other than one do not meet the criteria for linear equations in one variable.

Identify the following as either an expression or equation.
$\frac{2m}{3}+8$

• A. Expression
• B. Equation

Answer: (A)

Identify the following as either an expression or equation.
$4(a-2)=21$

• A. Expression
• B. Equation

Answer: (B)

Which of the following is a linear equation in one variable?

• A. $x+5=12$
• B. $x^2=25$
• C. $y+z=10$
• D. $x-3<7$

Answer: (A)

Which of the following is a linear equation in one variable?

• A. $2\left(x+5\right)-3x=x^2+1$
• B. $x\left(x+4\right)=0$
• C. $4\left(x+3\right)=2x+18$
• D. $x^2+5x=10$

Answer: (C)

Verify that the given value is a solution to the equation.
$y=-2;5y+4=14$

• A. Solution
• B. Not a solution

Answer: (B)

Verify that the given value is a solution to the equation.
$a=2;4a+3=a+9$

• A. Solution
• B. Not a solution

Answer: (A)

Addition and Subtraction Properties of Equality

A linear equation is an algebraic statement where the variable is to the first power, and solving it means finding the value of the variable that makes the equation true. For example, in the equation x − 6 = 0, the solution is the value of x that satisfies this equality.

To solve linear equations, the primary goal is to isolate the variable on one side of the equation. This is achieved by performing operations such as addition or subtraction on both sides of the equation, ensuring the balance or equality is maintained. This concept is known as creating equivalent equations, which have the same solution set.

The addition property of equality states that if a = b, then adding the same value c to both sides keeps the equation balanced:
\( a = b \implies a + c = b + c \).

Visualizing this with a balance scale helps: if you add weight to one side, the scale tips, but adding the same weight to the other side restores balance. This property is especially useful when an equation involves subtraction, as adding the opposite value cancels it out. For instance, in x − 6 = 0, adding 6 to both sides cancels the −6, isolating x:

\(x - 6 + 6 = 0 + 6\)
\(x = 6\)

To verify the solution, substitute x = 6 back into the original equation:
\$6 - 6 = 0\(
\)0 = 0\(, which confirms the solution is correct.

Similarly, the subtraction property of equality states that if a = b, then subtracting the same value c from both sides preserves equality:
\) a = b \implies a - c = b - c \(.

This property is useful when an equation contains addition, allowing subtraction to cancel it out. For example, in the equation 0 = x + 2, subtracting 2 from both sides isolates x:

\)0 - 2 = x + 2 - 2\(
\)-2 = x$

Checking the solution by substituting x = -2 back into the original equation yields a true statement, confirming the solution.

When solving linear equations, the result should always isolate the variable, either as x = number or number = x. The key is that the variable stands alone on one side.

In summary, the addition and subtraction properties of equality are fundamental tools for solving linear equations. The addition property helps eliminate subtraction by adding the opposite value to both sides, while the subtraction property cancels addition by subtracting the same value from both sides. Mastery of these properties enables solving a wide range of linear equations efficiently and accurately.

Addition and Subtraction Properties of Equality Example 2

To solve the linear equation y − 1.2 = 5.8, the goal is to isolate the variable y on one side. Since 1.2 is being subtracted from y, you can eliminate this by adding 1.2 to both sides of the equation. This maintains the equality because whatever operation you perform on one side must also be performed on the other.

Instead of rewriting the entire equation horizontally, you can perform the addition vertically for efficiency. Adding 1.2 to the left side cancels out the −1.2, leaving just y. On the right side, adding 1.2 to 5.8 results in 7.0. Thus, the solution is y = 7.

To verify the solution, substitute y = 7 back into the original equation:
\$7 - 1.2 = 5.8$.
Since this simplifies to a true statement, the solution is confirmed correct.

This method highlights the principle of maintaining balance in an equation by performing identical operations on both sides, a fundamental concept in solving linear equations. Using vertical addition or subtraction can streamline the process, making it easier to solve and check solutions efficiently.

Solve the given linear equation using addition and subtraction properties of equality.
$x+\frac{2}{8}=-\frac{3}{8}$

• A. $x=\frac58$
• B. $x=-\frac58$
• C. $x=5$
• D. $x=-5$

Answer: (B)

Solve the given linear equation using addition and subtraction properties of equality.
$-5.4+c=1.6$

• A. $c=1.6$
• B. $c=-1.6$
• C. $c=7.0$
• D. $c=-7.0$

Answer: (C)

Solve the given linear equation using addition and subtraction properties of equality.
$6h-(-12)=5+5h$

• A. $h=-17$
• B. $h=17$
• C. $h=7$
• D. $h=-7$

Answer: (D)

Solve the given linear equation using addition and subtraction properties of equality.
$2(x+5)=3(x-1)$

• A. $x=-13$
• B. $x=13$
• C. $x=3$
• D. $x=-3$

Answer: (B)

Solve the given linear equation using addition and subtraction properties of equality.
$3(y+3)+(1-y)=3y+13$

• A. $y=-4$
• B. $y=4$
• C. $y=0$
• D. $y=13$

Answer: (A)

Addition and Subtraction Properties of Equality Example 3

To translate the statement "a number decreased by seven is equal to 15" into a linear equation, start by representing the unknown number with a variable, commonly x. The phrase "decreased by seven" indicates subtraction, so the expression becomes x - 7. Since this is equal to 15, the equation is written as:

\[x - 7 = 15\]

To solve for x, isolate the variable by undoing the subtraction. Add 7 to both sides of the equation to maintain balance, resulting in:

\[x - 7 + 7 = 15 + 7\]

which simplifies to

\[x = 22\]

This means the number is 22. To verify the solution, substitute 22 back into the original equation:

\[22 - 7 = 15\]

Since this is a true statement, the solution is confirmed. This process demonstrates how to translate verbal expressions into linear equations and solve them by applying inverse operations, a fundamental skill in algebra.

Here’s what students ask on this topic:

What is the Addition Property of Equality and how is it used to solve linear equations?

The Addition Property of Equality states that if two expressions are equal, adding the same value to both sides keeps the equation balanced. Mathematically, if $a=b$, then $a+c=b+c$. This property is used to solve linear equations by isolating the variable. For example, in the equation $x-6=0$, adding 6 to both sides cancels the subtraction, giving $x=6$. This keeps the equation balanced and helps find the solution. Always perform the same operation on both sides to maintain equality.

How does the Subtraction Property of Equality help in solving linear equations?

The Subtraction Property of Equality states that if two expressions are equal, subtracting the same value from both sides keeps the equation balanced. Formally, if $a=b$, then $a-c=b-c$. This property is useful when an equation has addition that needs to be canceled to isolate the variable. For example, in $0=x+2$, subtracting 2 from both sides gives $0-2=x$ or $-2=x$. This isolates $x$ and helps find the solution.

What does it mean to isolate the variable in a linear equation?

Isolating the variable means rewriting the equation so that the variable stands alone on one side of the equation. This is the goal when solving linear equations because it allows us to find the exact value of the variable. For example, in the equation $x-6=0$, isolating $x$ involves adding 6 to both sides, resulting in $x=6$. The variable is now by itself, making it easy to identify the solution.

How can you verify the solution to a linear equation?

To verify a solution, substitute the found value back into the original equation and check if the equation holds true. For example, if the solution to $x-6=0$ is $x=6$, substitute 6 for $x$: $6-6=0$. Since $0=0$ is true, the solution is correct. This step confirms that the value satisfies the equation.

What is a linear equation and what does it mean for it to be first degree?

A linear equation is an equation that can be written in the form $ax+b=c$, where $a$, $b$, and $c$ are real numbers and $a\backslash neq\; 0$. The variable term has an exponent of one, which is why it is called first degree. This means the variable is not squared or raised to any higher power. Linear equations graph as straight lines and are foundational in algebra.

Why must operations be performed on both sides of an equation when solving it?

Operations must be performed on both sides of an equation to maintain equality and balance. An equation is like a balance scale; if you add, subtract, multiply, or divide one side, you must do the same to the other side to keep it balanced. For example, if you add 5 to one side, adding 5 to the other side ensures the two sides remain equal. This principle is essential for correctly solving equations without changing their solutions.