Problem 48 - Prealgebra

Question

Solve each equation. Check your solution. $$2(3+x)=14$$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Distribute the 2

Start by distributing the 2 on the left-hand side of the equation. Multiply both 3 and \(x\) by 2: \[ 2 imes 3 + 2 imes x = 6 + 2x \]Thus, the equation becomes:\[ 6 + 2x = 14 \]

Step 2: Isolate the term with the variable

Subtract 6 from both sides to isolate the term that contains \(x\):\[ 6 + 2x - 6 = 14 - 6 \]This simplifies to:\[ 2x = 8 \]

Step 3: Solve for x

Divide both sides by 2 to solve for \(x\):\[ \frac{2x}{2} = \frac{8}{2} \]This simplifies to:\[ x = 4 \]

Step 4: Check the solution

Substitute \(x = 4\) back into the original equation to check:\[ 2(3+4) = 14 \]Calculate inside the parentheses:\[ 2 imes 7 = 14 \]And confirm:\[ 14 = 14 \]The original equation holds true, so \(x = 4\) is correct.

Key Concepts

Understanding the Distributive Property

The distributive property is a foundational concept that's crucial when solving linear equations. It allows you to simplify expressions by "distributing" a multiplier across terms inside parentheses. For an equation like \(2(3+x)=14\), this means multiplying each term inside the parentheses by 2.

In this example, both "3" and "x" inside the parentheses are multiplied: \(2 \times 3 + 2 \times x = 6 + 2x\). This step eliminates the parentheses and prepares the equation for further simplification.

• The distributive property helps break down complex equations.
• It involves exact multiplication of all inside terms.
• Necessary for moving forward in solving equations.

Recognizing when to apply the distributive property can make solving equations more manageable, especially when dealing with positive, negative, or even multiple variables.

Grasping the Role of Variables

Variables are symbols, often letters like \(x\), used to represent unknown values in an equation. In the equation \(2(3+x)=14\), "\(x\)" is the variable, holding the place for an unknown number we need to find. Understanding variables is key to developing problem-solving skills.

Variables can stand for any number and are often used in equations to abstractly represent different possible values. This abstraction allows equations to express general rules:

• Variables are placeholders for numbers.
• Knowing the role of variables aids in creating equations.
• Helps illustrate how changes in numbers affect outcomes.

Mastering how to manipulate and solve for variables is fundamental in mathematics and forms the basis for exploring more complex algebraic concepts.

Isolating Variables: The Key to Solving Equations

Isolating the variable means getting the variable by itself on one side of the equation. It’s an essential step for solving equations, as it allows you to determine the variable's value.

In our example, after using the distributive property to simplify \(2(3+x)=14\) into \(6+2x=14\), the next step is to isolate \(x\). This involves moving the constant out of the way by subtracting 6 from both sides, simplifying the equation to \(2x=8\).

• Isolate variables by performing inverse operations.
• Ensure the variable remains on one side of the equation.
• Maintain equation balance.

Once you've isolated the variable, you'll usually need one final step to solve for it, such as division or multiplication.

Checking Solutions for Accuracy

Checking the solution of an equation is a vital step to ensure that the solved value is correct. After solving for the variable, it's helpful to substitute the value back into the original equation.

Let's verify \(x = 4\) by substituting back into the equation: \(2(3+4)=14\), calculating inside the parentheses gives \(2 \times 7 = 14\). Finally, needless to say, \(14=14\) verifies our solution is accurate.

• Substitution checks confirm solution correctness.
• Ensure both sides of the equation remain equal.
• Eliminates mistakes made in previous steps.

Always double-check your solution: it’s a good habit that helps catch errors early and increases confidence in your mathematical abilities.