Problem 83 - Intermediate Algebra

Question

Solve each equation. See Sections 2.1 and 5.8. $$ 2 x-7=3(x-4) $$

Step-by-Step Solution

Answer

The solution is \(x = 5\).

Step 1: Distribute Right Side

Start by expanding the expression on the right side of the equation. Multiply through the parentheses:\[ 3(x - 4) = 3x - 12 \]This gives us the equation:\[ 2x - 7 = 3x - 12 \]

Step 2: Simplify the Equation

Next, we will rearrange the equation to bring like terms to the same side. Subtract \(3x\) from both sides to move the \(x\)-terms to one side of the equation:\[ 2x - 3x - 7 = -12 \]This simplifies to:\[ -x - 7 = -12 \]

Step 3: Solve for x

Now, isolate \(x\) by adding 7 to both sides:\[ -x = -12 + 7 \]This simplifies to:\[ -x = -5 \]Multiply both sides by \(-1\) to solve for \(x\):\[ x = 5 \]

Step 4: Verify the Solution

Substitute \(x = 5\) back into the original equation to verify:\[ 2(5) - 7 = 3(5 - 4) \]Calculate both sides:\[ 10 - 7 = 3 \(5 - 4\) = 3(1) \]Both sides equal 3, so \(x = 5\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra, where we find the value of a variable that makes an equation true. When solving simple linear equations like the one in our example, the process involves a few systematic steps.
To solve the equation, we desire to get the variable, say \(x\), alone on one side. This is called isolating the variable. By performing the same operations on both sides of the equation, such as adding, subtracting, multiplying, or dividing, we preserve the balance of the equation.
Think of it like a balanced scale; if you add or take away something from one side, you must do the same to the other to maintain the balance. In every step, our goal is to simplify the equation until \(x\) stands alone.

Distributive Property

The distributive property is a useful algebra principle that allows us to simplify expressions involving parentheses. It states that \(a(b + c) = ab + ac\). This property lets us "distribute" multiplication over addition or subtraction inside the parentheses.
In our problem, utilizing the distributive property is crucial to move from \(3(x - 4)\) to \(3x - 12\). Here, we multiply the 3 by each term inside the parentheses separately. This expansion is crucial in preparing the equation for further simplification, as it removes the parentheses and sets the stage for combining like terms.

Combining Like Terms

Combining like terms is an essential step to simplify equations. Like terms are terms that have the same variable raised to the same power.
For example, in our equation \(2x - 7 = 3x - 12\), both \(2x\) and \(3x\) are like terms because they are both terms with the variable \(x\). By moving terms from one side of the equation to the other, we collect them together, enabling us to perform arithmetic operations on them.
Subtracting \(3x\) from both sides in our problem gathers all the \(x\) terms together. Similarly, constants like \(-7\) and \(-12\) are combined through addition or subtraction. By doing this, we simplify the problem further, making it easier to isolate the variable.

Verification of Solutions

Verification of solutions is the final step in solving equations to ensure the solution is correct. It involves substituting the found value back into the original equation.
In our example, after solving for \(x\) and finding \(x = 5\), we substitute 5 back into both sides of the original equation \(2x - 7 = 3(x-4)\).
By calculating each side separately and confirming that both are equal, we verify that our solution makes the original equation true. Verification is crucial because it confirms that no mistakes were made in the algebraic process, ensuring the validity of our answer.