Problem 40 - Elementary Algebra

Question

Solve. $$ 4 x-3=21 $$

Step-by-Step Solution

Answer

The solution is \( x = 6 \).

Step 1: Understand the Equation

The equation given is a simple linear equation: \( 4x - 3 = 21 \). The main goal is to find the value of \( x \) that makes this equation true.

Step 2: Isolate the Term Involving x

First, we need to move the constant term on the left side to the right side of the equation. To do this, add 3 to both sides of the equation to cancel out the \(-3\):\[4x - 3 + 3 = 21 + 3\]This simplifies to:\[4x = 24\]

Step 3: Solve for x

Now, we need to isolate \( x \) by dividing both sides of the equation by 4:\[\frac{4x}{4} = \frac{24}{4}\]Simplifying this gives:\[x = 6\]

Step 4: Verify the Solution

To ensure that the solution is correct, substitute \( x = 6 \) back into the original equation and check:\[4(6) - 3 = 24 - 3 = 21\]Since both sides of the equation are equal, \( x = 6 \) is the correct solution.

Key Concepts

Isolation of Variables

When it comes to solving linear equations, one of the crucial steps is the **isolation of variables**. This means rearranging the equation so that the variable you're solving for is by itself on one side of the equation. In our example equation, \(4x - 3 = 21\), our goal is to isolate \(x\). This involves moving any other numbers or terms to the other side. First, we look at the equation and see that there is a \(-3\) attached to \(4x\). By adding \(3\) to both sides, we effectively "cancel out" the \(-3\), thus simplifying to \(4x = 24\). Now, \(x\) is closer to being by itself. The final step to isolate \(x\) is to eliminate any coefficients attached to it. Here, \(x\) is multiplied by \(4\), so we divide both sides by \(4\). This leaves us with the equation \(x = 6\). By isolating the variable, we easily find its value and solve the equation.

Verifying Solutions

After solving an equation, it's important to make sure the solution is correct. This is where **verifying solutions** comes into play. Verification is simply checking if the value you found actually satisfies the original equation. In our example, we found \(x = 6\). To verify this, substitute \(6\) back into the original equation \(4x - 3 = 21\). - Multiply \(4\) by \(6\) to get \(24\).- Subtract \(3\) to end up with \(21\).The result, \(21\), matches the right side of the original equation. This confirms that our solution, \(x = 6\), is indeed correct. Verification adds confidence to your solution and ensures no mistakes were made during the solving process.

Substitution Method

The **substitution method** is another useful tool in mathematics, especially for verifying solutions or solving systems of equations. In our case, we use substitution to verify that our solution is correct.By substituting a found value back into the original equation, you recheck the work. If we revisit our solution \(x = 6\), insert it back into \(4x - 3 = 21\) which becomes \(4(6) - 3\). - Calculate \(4 \times 6 = 24\)- Subtract \(3\) to get \(21\)The equation balances which confirms that \(x = 6\) is the correct value.Substitution is particularly powerful because:

• It's straightforward and works well for verifying solutions.
• It can support isolating strategies in complex problems.

Using the substitution method provides an extra layer of assurance in your solutions.