The Multiplication and Division Properties of Equality: Videos & Practice Problems

Solving linear equations involves using properties of equality—addition, subtraction, multiplication, and division—to isolate variables. When variables have fractional coefficients, multiply both sides by the reciprocal to simplify, leveraging the concept that a number times its reciprocal equals one. For example, to solve $\mathrm{3/4}x=9$, multiply both sides by $\mathrm{4/3}$ to isolate $x$. This approach maintains equation balance, akin to a scale, ensuring equivalent expressions. Mastery of these techniques is essential for handling polynomials, coefficients, and terms in algebraic expressions.

Multiplication and Division Properties of Equality

When solving linear equations, it is essential to maintain balance by performing the same operation on both sides of the equation. This principle ensures that the equality remains true throughout the process. Beyond addition and subtraction, multiplication and division properties of equality are powerful tools used to isolate variables and solve equations effectively.

The multiplication property of equality states that if two expressions are equal, multiplying both sides by the same nonzero number preserves the equality. Formally, if \(a = b\), then \(a \times c = b \times c\). This property is particularly useful when an equation involves division, as multiplication can cancel out the division. For example, in the equation \(\frac{x}{2} = 9\), multiplying both sides by 2 eliminates the denominator, yielding \(x = 9 \times 2 = 18\). This step isolates the variable \(x\) and solves the equation.

Similarly, the division property of equality states that if \(a = b\), then dividing both sides by the same nonzero number \(c\) maintains equality: \(\frac{a}{c} = \frac{b}{c}\). This property is especially helpful when the variable is multiplied by a coefficient. For instance, in \$20 = 5x\(, dividing both sides by 5 cancels the multiplication, resulting in \)x = \frac{20}{5} = 4\(. This isolates \)x\( and provides the solution.

When solving more complex linear equations, it is important to first use addition or subtraction to isolate the variable term before applying multiplication or division to solve for the variable itself. For example, consider the equation \)3a - 4 = 11\(. To isolate the term with the variable, add 4 to both sides to get \)3a = 15\(. Then, divide both sides by 3 to solve for \)a\(: \)a = \frac{15}{3} = 5\(.

Checking solutions by substituting the found value back into the original equation confirms their correctness. For \)a = 5\(, substituting into \)3a - 4 = 11\( gives \)3 \times 5 - 4 = 15 - 4 = 11$, which is true, verifying the solution.

Mastering these properties of equality—addition, subtraction, multiplication, and division—enables efficient solving of linear equations by systematically isolating variables while maintaining equation balance. Understanding and applying these inverse operations are fundamental skills in algebra that support solving increasingly complex problems.

Multiplication and Division Properties of Equality Example 1

To solve the linear equation 1.2a + 2.3 = 5.9, the goal is to isolate the variable a. Begin by using the subtraction property of equality to eliminate the constant term on the left side. Subtract 2.3 from both sides to maintain balance, resulting in 1.2a = 5.9 - 2.3. Simplifying the right side gives 1.2a = 3.6. Next, apply the division property of equality by dividing both sides by 1.2 to solve for a. This yields a = \frac{3.6}{1.2} = 3.

To verify the solution, substitute a = 3 back into the original equation. Calculate 1.2 \times 3 + 2.3, which equals 3.6 + 2.3 = 5.9. Since both sides of the equation are equal, the solution is confirmed as correct.

This process highlights the importance of using properties of equality—subtraction and division—to isolate variables in linear equations. Checking solutions by substitution ensures accuracy and reinforces understanding of equation balance. Mastery of these steps is essential for solving and verifying linear equations efficiently.

Solve the given linear equation using multiplication and division properties of equality.
$-8x=64$

• A. $x=64$
• B. $x=-64$
• C. $x=-8$
• D. $x=8$

Answer: (C)

Solve the given linear equation using multiplication and division properties of equality.
$\frac{y}{4}=-\frac{21}{6}$

• A. $y=14$
• B. $y=-14$
• C. $y=7$
• D. $y=-7$

Answer: (B)

Solve the given linear equation using multiplication and division properties of equality.
$-\frac{126}{14}=3y$

• A. $y=27$
• B. $y=-27$
• C. $y=3$
• D. $y=-3$

Answer: (D)

Solve the given linear equation, then check your answer
$-(h+3)=11$

• A. $h=-14$
• B. $h=-\frac{11}{3}$
• C. $h=-33$
• D. $h=33$

Answer: (A)

Solve the given linear equation, then check your answer
$0.5t+1.5t=7+3-4$

• A. $t=1$
• B. $t=-6$
• C. $t=3$
• D. $t=-3$

Answer: (C)

Multiplication and Division Properties of Equality Example 2

To translate the statement "three times L equals 54" into a linear equation, recognize that "three times" indicates multiplication. This means the expression can be written as \$3L = 54\(, where \)L\( is the variable representing the unknown value. The equal sign denotes equality between the two sides of the equation.

To solve for \)L\(, apply the division property of equality. Since \)L\( is multiplied by 3, divide both sides of the equation by 3 to isolate \)L\(. This gives:

\[L = \frac{54}{3}\]

Calculating the right side, \)54 \div 3 = 18\(, so:

\[L = 18\]

To verify the solution, substitute \)L = 18\( back into the original equation:

\[3 \times 18 = 54\]

Since \)54 = 54\( is a true statement, the solution \)L = 18$ is correct. This process demonstrates how to translate verbal expressions into algebraic equations and solve them using inverse operations, reinforcing the fundamental concept of maintaining equality while isolating variables.

Multiplication Property for Fraction Coefficients

When solving linear equations where the variable has a fractional coefficient, such as \(\frac{3}{4}x = 9\), the key strategy is to eliminate the fraction by multiplying both sides of the equation by the reciprocal of the fraction. The reciprocal of a fraction is obtained by flipping its numerator and denominator. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Multiplying a fraction by its reciprocal always results in 1, which helps isolate the variable.

Applying this to the equation \(\frac{3}{4}x = 9\), multiply both sides by \(\frac{4}{3}\):

\[\frac{4}{3} \times \frac{3}{4} x = \frac{4}{3} \times 9\]

The left side simplifies to \$1 \times x = x\(, and the right side becomes:

\[\frac{4}{3} \times 9 = \frac{4}{3} \times \frac{9}{1} = \frac{4 \times 9}{3 \times 1} = \frac{36}{3} = 12\]

Thus, the solution is \)x = 12\(. This method maintains the balance of the equation by applying the multiplication property of equality, which states that multiplying both sides of an equation by the same nonzero number does not change the equality.

Another example is solving \)10 = \frac{5}{3}y\(. To isolate \)y\(, multiply both sides by the reciprocal of \)\frac{5}{3}\(, which is \)\frac{3}{5}\(:

\[\frac{3}{5} \times 10 = \frac{3}{5} \times \frac{5}{3} y\]

Simplifying the right side gives \)1 \times y = y\(, and the left side becomes:

\[\frac{3}{5} \times 10 = \frac{3}{5} \times \frac{10}{1} = \frac{3 \times 10}{5 \times 1} = \frac{30}{5} = 6\]

Therefore, \)y = 6$.

This approach to solving linear equations with fractional coefficients is straightforward and effective. By multiplying both sides by the reciprocal of the fraction, the variable is isolated, allowing for easy simplification and solution. This technique reinforces understanding of the multiplication property of equality and the concept of reciprocals, which are fundamental in algebraic manipulation.

Solve the given linear equation.
$\frac{2}{3}x=10$

• A. $x=5$
• B. $x=10$
• C. $x=\frac{20}{3}$
• D. $x=15$

Answer: (D)

Solve the given linear equation.
$-6=\frac{3}{5}t$

• A. $t=10$
• B. $t=-3.6$
• C. $t=3.6$
• D. $t=-10$

Answer: (D)

Solve the given linear equation, then check your answer.
$\frac{1}{2}=\frac{3}{4}x$

• A. $x=\frac{3}{8}$
• B. $x=2$
• C. $x=\frac{2}{3}$
• D. $x=\frac{1}{3}$

Answer: (C)

Here’s what students ask on this topic:

What is the multiplication property of equality and how is it used to solve linear equations?

The multiplication property of equality states that if two values are equal, multiplying both sides of the equation by the same nonzero number keeps the equation balanced. In MathML, this is expressed as: $a=b\to a\cdot c=b\cdot c$. This property is especially useful when the variable is divided by a number. For example, to solve $\frac{x}{2}=9$, multiply both sides by 2 to cancel the division, resulting in $x=18$. This isolates the variable, allowing you to find its value. Always perform the same operation on both sides to maintain equality.

How does the division property of equality help in solving equations with multiplication?

The division property of equality states that if two values are equal, dividing both sides by the same nonzero number keeps the equation balanced. In MathML: $a=b\to \frac{a}{c}=\frac{b}{c}$. This property is used when the variable is multiplied by a coefficient. For example, in $20=5x$, dividing both sides by 5 cancels the multiplication, giving $x=4$. This isolates the variable. Remember, always divide both sides by the same number to maintain equality and solve correctly.

How do you solve linear equations with fractional coefficients using the multiplication property of equality?

When a variable has a fractional coefficient, multiply both sides of the equation by the reciprocal of that fraction to isolate the variable. The reciprocal of a fraction $\frac{a}{b}$ is $\frac{b}{a}$. For example, to solve $\frac{3}{4}x=9$, multiply both sides by $\frac{4}{3}$. This cancels the fraction on the left, leaving $x=9\cdot \frac{4}{3}$. Simplify the right side to get $x=12$. This method ensures the variable is isolated correctly even with fractional coefficients.

Why is it important to perform the same operation on both sides of an equation when solving linear equations?

Performing the same operation on both sides of an equation maintains the balance and equality of the equation, similar to keeping a scale balanced. This principle ensures that the value of the equation does not change while isolating the variable. For example, if you add, subtract, multiply, or divide one side by a number, you must do the same to the other side. This is the foundation of the properties of equality, including addition, subtraction, multiplication, and division properties. Without this, the equation would no longer represent the same relationship, leading to incorrect solutions.

How do you check the solution of a linear equation after solving it using multiplication or division properties?

To check the solution, substitute the found value of the variable back into the original equation. Then simplify both sides to verify if they are equal. For example, if you solve $\frac{x}{2}=9$ and find $x=18$, substitute 18 for x: $\frac{18}{2}=9$. Simplify the left side to get 9, which equals the right side. This confirms the solution is correct. Checking ensures no mistakes were made during solving.