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 these equations involves isolating the variable using properties of equality, such as adding or subtracting the same value from both sides to maintain balance. The solution is the value that satisfies the equation, forming the solution set. Understanding these concepts is foundational for working with polynomials, exponents, and algebraic expressions in standard form.

Introduction to Linear Equations

An equation is a mathematical statement asserting that two algebraic expressions are equal. A specific type of equation, known as a linear equation, can be expressed in the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are real numbers and \(a \neq 0\). This condition ensures the presence of a variable term, which is essential for the equation to be linear. The variable, commonly denoted as \(x\), can actually be any letter, and the term "linear" or "first degree equation" refers to the fact that the variable is raised to the power of one, indicating the equation's degree.

The solution to a linear equation is the value of the variable that makes the equation true when substituted back into it. For example, consider the equation \$2x + 6 = 0\(. To determine if \)x = -3\( is a solution, substitute \)-3\( for \)x\(:

\[2(-3) + 6 = -6 + 6 = 0\]

Since the left side equals the right side, \)x = -3\( is indeed a solution. Conversely, for the equation \)5 = 8w - 3\(, substituting \)w = -1\( yields:

\[8(-1) - 3 = -8 - 3 = -11\]

Since \)5 \neq -11\(, \)w = -1\( is not a solution. This illustrates the importance of verifying potential solutions by substitution.

The solution set of a linear equation is the collection of all values that satisfy the equation, typically expressed using curly brackets. For instance, the solution set for \)2x + 6 = 0\( with solution \)x = -3\( is written as \)\{-3\}$. Understanding how to identify and verify solutions is fundamental before progressing to solving linear equations algebraically.

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, it is only raised to the first power, which is a key characteristic of linear equations. Combining like terms simplifies this to 4y = 2, clearly showing it is a linear equation in one variable.

In contrast, an equation like 6x + 1 = 2t² is not a linear equation in one variable because it contains two different variables, x and t, and the variable t is squared. Linear equations must have only one variable raised to the first power, so the presence of multiple variables and exponents other than one disqualifies this expression.

In summary, a linear equation in one variable must have exactly one variable raised to the power of one and must be an equation with an equal sign. Expressions without an equal sign, with multiple variables, or with 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
• C. Both A and B
• D. None of the above

Answer: (A)

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

• A. Expression
• B. Equation
• C. Both A and B
• D. None of the above

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.

Answer: (A)

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

• A. $2(x+5)-3x=x^2+1$
• B. $x(x+4)=0$
• C. $4(x+3)=2x+18$
• 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
• C. $y=-2$ is a solution for some values of $y$

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
• C. $a=2$ is not a valid 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 creation of equivalent equations—equations that maintain the same balance or truth value. This concept is often illustrated using the balance scale analogy: if you add or remove the same weight from both sides, the scale remains balanced, just as applying the same operation to both sides of an equation preserves equality.

The addition property of equality states that if a = b, then adding the same value c to both sides results in a + c = b + c. This property is particularly useful when an equation involves subtraction, as adding the opposite value cancels out the subtraction. For instance, in x − 6 = 0, adding 6 to both sides cancels the −6, isolating x and yielding x = 6. This solution can be verified by substituting back into the original equation: 6 − 6 = 0, which simplifies to 0 = 0, confirming the correctness.

Similarly, the subtraction property of equality states that if a = b, then subtracting the same value c from both sides results in a − c = b − c. This property is used to eliminate addition in an equation. For example, in 0 = x + 2, subtracting 2 from both sides cancels the +2, isolating x and giving x = −2. Checking this solution by substitution confirms its validity.

When solving linear equations, the solution should always isolate the variable, either in the form x = number or number = x. The key is that the variable stands alone on one side of the equation. By applying the addition and subtraction properties of equality appropriately, one can systematically solve linear equations by canceling out unwanted terms and isolating the variable.

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 side.

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

\[ y = 7 \]

To verify this solution, substitute y = 7 back into the original equation:

\[ 7 - 1.2 = 5.8 \]

Since the left side equals the right side, the solution is confirmed as correct. This method of solving linear equations by isolating the variable and checking the solution ensures accuracy and reinforces understanding of equality principles in algebra.

Solve the given linear equation using addition and subtraction properties of equality.
$m-9=-6$

• A. $m=-3$
• B. $m=9$
• C. $m=3$
• D. $m=-6$

Answer: (C)

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

• A. $x=5$
• B. $x=7$
• C. $x=-7$
• D. $x=12$

Answer: (B)

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

• A. $x=\frac{5}{8}$
• B. $x=-\frac{5}{8}$
• 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)

Addition and Subtraction Properties of Equality Example 3

To solve the linear equation 4x + 7 = 3x + 10, the goal is to isolate the variable x on one side and constants on the other. Since x appears on both sides, we use the properties of equality, specifically addition and subtraction, to move terms and simplify the equation.

Start by moving all terms containing x to one side. Subtract 3x from both sides to eliminate it from the right side:

\$4x + 7 - 3x = 3x + 10 - 3x\(

This simplifies to:

\)x + 7 = 10\(

Next, isolate x by subtracting 7 from both sides:

\)x + 7 - 7 = 10 - 7\(

Which simplifies to:

\)x = 3\(

To verify the solution, substitute x = 3 back into the original equation:

\)4(3) + 7 = 3(3) + 10\(

Calculate each side:

\)12 + 7 = 9 + 10\(

\)19 = 19$

Since both sides are equal, the solution x = 3 is correct. This process demonstrates how to solve linear equations with variables on both sides by combining like terms and maintaining equality through balanced operations.

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.
$3(y+3)+(1-y)=3y+14$

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

Answer: (A)

Addition and Subtraction Properties of Equality Example 4

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:

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

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 eliminating subtraction. For example, in the equation $x-6=0$, adding 6 to both sides cancels the -6, isolating $x$. This results in $x=6$, the solution. Using this property ensures the equation remains equivalent, allowing us to find the value of the variable that makes the equation true.

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 for eliminating addition in equations. For example, in $0=x+2$, subtracting 2 from both sides cancels the +2, isolating $x$. The result is $x=-2$. This method helps isolate the variable, making it easier to solve linear equations while maintaining equality.

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

Isolating the variable means rearranging 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 $x=6$. This process uses properties of equality, such as addition or subtraction, to maintain balance and find the solution that makes the equation true.

How can you verify if a value is a solution to a linear equation?

To verify if a value is a solution, substitute it into the original equation in place of the variable and check if the equation holds true. For example, for the equation $2x+6=0$, if we test $x=-3$, substitute to get $2(-3)+6=0$. Simplifying, $-6+6=0$, which is true. Since the equation is satisfied, $x=-3$ is a solution. If the substitution does not result in a true statement, the value is not a solution.

What is a linear equation and how is it different from other types of equations?

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 $x$ has an exponent of 1, making it a first-degree polynomial. This contrasts with quadratic or higher-degree equations where variables have exponents greater than 1. Linear equations graph as straight lines and are solved by isolating the variable using properties of equality, such as addition and subtraction, to maintain balance.

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 represents two expressions that are equal, like a balanced scale. If you add, subtract, multiply, or divide only one side, the balance is disturbed, and the equation no longer holds true. For example, if $a=b$, adding $c$ to both sides results in $a+c=b+c$, preserving equality. This principle is fundamental when using the addition and subtraction properties of equality to isolate variables and solve linear equations.