Problem 28 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 4 x+7=-17 $$

Step-by-Step Solution

Answer

The solution is \( x = -6 \).

Step 1: Isolate the Variable Term

First, we want to get the term with the variable \( x \) by itself. To do this, subtract 7 from both sides of the equation.\[ 4x + 7 - 7 = -17 - 7 \] Which simplifies to \[ 4x = -24 \]

Step 2: Solve for the Variable

Next, divide both sides of the equation by 4 to solve for \( x \).\[ \frac{4x}{4} = \frac{-24}{4} \] Which gives us \[ x = -6 \]

Step 3: Verify the Solution

Substitute \( x = -6 \) back into the original equation to verify the solution is correct.\[ 4(-6) + 7 \] Calculate \[ = -24 + 7 \] \[ = -17 \] Since the original equation \( 4x + 7 = -17 \) holds true, the solution \( x = -6 \) is correct.

Key Concepts

Isolate the Variable

When solving linear equations, such as \( 4x + 7 = -17 \), one of the crucial first steps is to "isolate the variable." This means we rearrange the equation in such a way that the variable term stands alone on one side of the equation. This helps us to directly find the value of the variable using basic operations.

• To isolate the variable \( x \), we begin by removing any constants added or subtracted to it from one side. In our example, the constant is \( 7 \).
• We subtract \( 7 \) from both sides of the equation to maintain balance (\( 4x + 7 - 7 = -17 - 7 \)).
• This will simplify the equation to \( 4x = -24 \), getting us closer to solving for the variable \( x \).

By effectively isolating the variable, we can then focus on eliminating any coefficients or additional factors that influence the variable.

Verify the Solution

Once you've found a possible solution, it's essential to "verify the solution" by substituting your answer back into the original equation. This ensures the solution makes the equation true. If substituting back gives you an unequal statement, there might have been an error in the process.

• In our example, we found \( x = -6 \). By substituting \( -6 \) back into the original equation \( 4x + 7 = -17 \), we perform the operation: \( 4(-6) + 7 \).
• Calculate: \( -24 + 7 = -17 \). Since this equals \( -17 \), the initial solution checks out.

Verifying your solution helps catch possible mistakes and confirm that the solution you obtained is indeed correct.

Simplify the Equation

Simplifying the equation is about making it easier to solve by reducing unnecessary complexity. This often involves combining like terms and performing basic arithmetic operations to clear up the equation.

• While isolating the variable, simplifying steps help maintain clarity. In the example, taking \( 4x + 7 = -17 \) to \( 4x = -24 \) simplifies by removing the constant \( 7 \).
• Then, further simplify by handling coefficients: \( \frac{4x}{4} = \frac{-24}{4} \).
• This leads directly to \( x = -6 \), obtained in the simplest form.

Simplification is key to ensuring you can straightforwardly solve for the variable without getting bogged down by unnecessarily complicated equations.