7 Ways to Solve Equations With Distributive Property

Unlocking the Power of Distributive Property: A Step-by-Step Guide

The distributive property is a fundamental concept in algebra that allows us to simplify complex equations by breaking them down into manageable parts. It states that for any numbers a, b, and c, a(b + c) = ab + ac. This powerful tool can help you solve a wide range of equations, from simple linear equations to more complex quadratic equations. In this article, we’ll explore seven ways to solve equations using the distributive property.

Method 1: Simple Linear Equations

The distributive property can be used to simplify linear equations by distributing a single term to multiple terms. For example, let’s consider the equation 2(x + 3) = 12.

📝 Note: When solving linear equations, always start by isolating the variable on one side of the equation.

To solve for x, we can use the distributive property to expand the left side of the equation:

2(x + 3) = 2x + 6

Now, we can simplify the equation by combining like terms:

2x + 6 = 12

Subtracting 6 from both sides gives us:

2x = 6

Dividing both sides by 2 gives us the final solution:

x = 3

Method 2: Multiplying Both Sides by a Constant

In some cases, we may need to multiply both sides of the equation by a constant to eliminate fractions or decimals. For example, let’s consider the equation ¹⁄₂(x + 2) = 3.

To eliminate the fraction, we can multiply both sides of the equation by 2:

2(¹⁄₂(x + 2)) = 2(3)

Using the distributive property, we can expand the left side of the equation:

x + 2 = 6

Subtracting 2 from both sides gives us:

x = 4

Method 3: Solving Quadratic Equations

The distributive property can also be used to solve quadratic equations. For example, let’s consider the equation x^2 + 5x + 6 = 0.

To factor the left side of the equation, we can use the distributive property to expand the expression:

x^2 + 5x + 6 = (x + 3)(x + 2) = 0

Now, we can set each factor equal to zero and solve for x:

x + 3 = 0 or x + 2 = 0

Solving for x gives us:

x = -3 or x = -2

Method 4: Using the Distributive Property with Multiple Variables

In some cases, we may need to use the distributive property with multiple variables. For example, let’s consider the equation 2(x + y) = 12.

To solve for x and y, we can use the distributive property to expand the left side of the equation:

2x + 2y = 12

Now, we can simplify the equation by combining like terms:

x + y = 6

To find the values of x and y, we can use substitution or elimination methods.

Method 5: Solving Systems of Equations

The distributive property can also be used to solve systems of equations. For example, let’s consider the system of equations:

x + y = 4 2(x + y) = 12

To solve the system, we can use the distributive property to expand the second equation:

2x + 2y = 12

Now, we can simplify the equation by combining like terms:

x + y = 6

Subtracting the first equation from the second equation gives us:

x = 2

Substituting x = 2 into the first equation gives us:

2 + y = 4

Solving for y gives us:

y = 2

Method 6: Using the Distributive Property with Fractions

In some cases, we may need to use the distributive property with fractions. For example, let’s consider the equation ¹⁄₂(x + 2) = ³⁄₄.

To eliminate the fractions, we can multiply both sides of the equation by 4:

2(x + 2) = 3

Using the distributive property, we can expand the left side of the equation:

2x + 4 = 3

Subtracting 4 from both sides gives us:

2x = -1

Dividing both sides by 2 gives us the final solution:

x = -¹⁄₂

Method 7: Solving Equations with Decimals

Finally, the distributive property can be used to solve equations with decimals. For example, let’s consider the equation 0.5(x + 2) = 3.

To eliminate the decimal, we can multiply both sides of the equation by 2:

x + 2 = 6

Subtracting 2 from both sides gives us:

x = 4

In conclusion, the distributive property is a powerful tool for solving equations. By using the distributive property, we can simplify complex equations and solve for unknown variables. Whether we’re solving simple linear equations or more complex quadratic equations, the distributive property can help us find the solution.

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