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. Multiplication and division properties help cancel coefficients, including fractional coefficients, by multiplying both sides by reciprocals, ensuring balanced equations. For example, to solve $3/4x=9$, multiply both sides by $4/3$ to isolate $x$. Understanding coefficients, terms, and operations is essential for mastering polynomial expressions and linear equations in standard form.

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 to isolate the variable. For example, in the equation \(\frac{x}{2} = 9\), multiplying both sides by 2 eliminates the denominator, resulting in \(x = 18\).

Conversely, the division property of equality asserts that dividing both sides of an equation by the same nonzero number maintains equality. If \(a = b\), then \(\frac{a}{c} = \frac{b}{c}\). This property is especially helpful when the variable is multiplied by a coefficient. For instance, in \$20 = 5x\(, dividing both sides by 5 isolates \)x\(, yielding \)x = 4\(.

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. Consider the equation \)3a - 4 = 11\(. Adding 4 to both sides removes the subtraction, giving \)3a = 15\(. Then, dividing both sides by 3 isolates \)a\(, resulting in \)a = 5\(.

Verifying solutions by substituting the found value back into the original equation confirms the correctness of the solution. For example, substituting \)a = 5\( into \)3a - 4 = 11\( yields \)3 \times 5 - 4 = 15 - 4 = 11$, confirming the solution is accurate.

Mastering these properties of equality—addition, subtraction, multiplication, and division—enables the systematic solving of linear equations by maintaining balance and creating equivalent equations. This foundational understanding is crucial for progressing to more advanced algebraic problem-solving.

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\( is the variable representing the unknown value. The goal is to determine the value of \)L\( that satisfies this equation.

To solve for \)L\(, apply the division property of equality, which states that if both sides of an equation are divided by the same nonzero number, the equality remains true. Since \)L\( is multiplied by 3, divide both sides of the equation by 3 to isolate \)L\(:

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

Simplifying both sides gives:

\[L = 18\]

To verify this 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 statements into algebraic equations and solve them using fundamental properties of equality.

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 simplifies the equation and isolates 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:

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

Thus, \)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 preserves 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:

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

Therefore, \)y = 6$. This approach effectively removes fractional coefficients, allowing for straightforward solutions to linear equations.

Understanding how to multiply both sides of an equation by the reciprocal of a fractional coefficient is essential for solving linear equations efficiently. This technique leverages the fundamental property that the product of a number and its reciprocal equals one, simplifying the equation and isolating the variable. By consistently applying this method, you can confidently solve linear equations involving fractional coefficients and verify solutions by substitution.

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. $x=\frac38$
• B. $x=2$
• C. $x=\frac23$
• D. $x=\frac13$

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 expressions are equal, multiplying both sides by the same nonzero value keeps the equality true. In MathML, this is written as: $a=b\to a\cdot c=b\cdot c$. This property is especially useful when solving linear equations that involve division. For example, if you have $\frac{x}{2}=9$, you can multiply both sides by 2 to cancel the division and isolate $x$. This keeps the equation balanced and helps find the solution efficiently. Always remember to perform the same operation on both sides to maintain equality.

How do you apply the division property of equality to solve equations with multiplication?

The division property of equality states that if two expressions are equal, dividing both sides by the same nonzero value keeps the equality true. In MathML, it is expressed as: $a=b\to \frac{a}{c}=\frac{b}{c}$. This property is used to solve equations where the variable is multiplied by a coefficient. For example, in $20=5x$, dividing both sides by 5 isolates $x$. This cancels the multiplication and maintains the equation's balance. Always divide both sides by the same number to keep the equality valid and solve for the variable.

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

When a linear equation has a fractional coefficient, such as $\frac{3}{4}x=9$, you can solve it by multiplying both sides by the reciprocal of the fraction. The reciprocal of $\frac{3}{4}$ is $\frac{4}{3}$. Multiplying both sides by this reciprocal cancels the fraction, isolating $x$. In MathML: $\frac{4}{3}\cdot \frac{3}{4}x=\frac{4}{3}\cdot 9$. This simplifies to $x=\frac{36}{3}$, which further reduces to $x=12$. This method maintains the equation's balance and helps solve equations with fractional coefficients effectively.

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 is crucial because it maintains the equality and balance of the equation. Think of the equation as a balanced scale; if you add, subtract, multiply, or divide one side, you must do the same to the other side to keep it balanced. This principle ensures that the value of the variable remains consistent and the equation remains true. For example, if you multiply one side by 3, multiplying the other side by 3 keeps the equation equivalent. This approach is fundamental when using the addition, subtraction, multiplication, and division properties of equality to isolate variables and solve linear equations accurately.

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

To check the solution of a linear equation, substitute the found value of the variable back into the original equation. For example, if you solved $\frac{x}{2}=9$ and found $x=18$, plug 18 back in: $\frac{18}{2}=9$. Simplify the left side to verify if it equals the right side. If both sides are equal, the solution is correct. This step confirms that the operations performed preserved the equality and that the solution satisfies the original equation.