Problem 18 - Algebra and Trigonometry

Question

The given equation is either linear or equivalent to a linear equation. Solve the equation. \(4 x+7=9 x-13\)

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Identify the Type of Equation

We observe that the equation provided is \(4x+7=9x-13\), which is a linear equation. Linear equations have the general form \(ax + b = cx + d\). This equation can be solved by isolating \(x\).

Step 2: Move Terms Involving x to One Side

Subtract \(4x\) from both sides of the equation to get all terms involving \(x\) on the right side: \(7 = 9x - 4x - 13\). This simplifies to \(7 = 5x - 13\).

Step 3: Move the Constant Term to the Other Side

Add \(13\) to both sides to bring the constants on the left side: \(7 + 13 = 5x\). This simplifies to \(20 = 5x\).

Step 4: Solve for x

Divide both sides by \(5\) to isolate \(x\): \(x = \frac{20}{5}\). Simplifying this gives \(x = 4\).

Step 5: Verify the Solution

Substitute \(x=4\) back into the original equation to verify: \(4(4) + 7 = 9(4) - 13\). Calculating both sides gives \(16 + 7 = 36 - 13\), which simplifies to \(23 = 23\). This confirms the solution is correct.

Key Concepts

Equation Solving

Solving linear equations involves finding the value of the variable that makes the equation true. A linear equation, like the one in this exercise, has variables only raised to the first power. To solve such an equation, we perform operations to one or both sides to rearrange and simplify it.
When solving, keep the equation balanced by performing the same operation on each side. This means:

• Adding the same number to both sides
• Subtracting the same number from both sides
• Multiplying both sides by the same non-zero number
• Dividing both sides by the same non-zero number

These steps help us to logically work towards a simpler equation where the variable is isolated and its value is clear.

Isolating Variables

Isolating the variable is a key step in solving any equation, especially linear equations. The goal is to have the variable appear by itself on one side of the equation, which makes it easier to find its value. In our example, we began by moving the terms involving the variable, in this case, terms with "x", to one side.
To do this:

• We subtracted smaller multiples of "x" from larger multiples on either side, resulting in one side solely having terms with "x".
• Moved constant terms to the opposite side by adding or subtracting them from both sides.

After these steps, dividing the remaining term by the coefficient of "x" gives its value. Isolating variables clarifies the relationship between terms and helps simplify calculations.

Verifying Solutions

Verification is a crucial step that ensures the correctness of any solution. After solving an equation, substitute the found value back into the original equation to check if both sides equal.
This simple act confirms:

• The calculated value is indeed correct.
• No mistakes were made during the solving process.
• The solution satisfies the original problem statement.

For example, by substituting the value of "x = 4" back into the original equation, both sides equaled 23, confirming our solution. This step not only boosts confidence in the answer but is also a fantastic way to prevent lingering doubts about accuracy.