Problem 32 - A Graphical Approach to College Algebra

Question

$$\text { Solve each equation analytically. Check it analytically, and then support your solution graphically.}$$ $$1.1 x-2.5=0.3(x-2)$$

Step-by-Step Solution

Answer

The solution is \(x \approx 2.714\).

Step 1: Distribute on the Right Side

Start by distributing the 0.3 on the right-hand side of the equation. The original equation is: \[ x - 2.5 = 0.3(x - 2) \]Applying the distributive property, we get: \[ x - 2.5 = 0.3x - 0.6 \]

Step 2: Combine Like Terms

Bring all the terms involving \(x\) to one side of the equation and the constant terms to the other. Subtract \(0.3x\) from both sides:\[ x - 0.3x - 2.5 = -0.6 \]Simplifying, we have:\[ 0.7x - 2.5 = -0.6 \]

Step 3: Isolate x

Now, move the constant term \(-2.5\) to the other side by adding 2.5 to both sides:\[ 0.7x = -0.6 + 2.5 \]Simplifying, we get: \[ 0.7x = 1.9 \]

Step 4: Solve for x

Divide both sides by 0.7 to solve for \(x\):\[ x = \frac{1.9}{0.7} \]Calculating gives:\[ x \approx 2.714 \]

Step 5: Check the Solution Analytically

Substitute \(x = 2.714\) back into the original equation: \[ x - 2.5 = 0.3(x-2) \] Substitute \(x\):\[ 2.714 - 2.5 \approx 0.3(2.714 - 2) \]Calculate each side:\[ 0.214 \approx 0.3 \times 0.714 \approx 0.214 \]Both sides are approximately equal, confirming the solution is correct.

Key Concepts

Distributive Property

The distributive property is a crucial tool in algebra for breaking down and simplifying equations, especially when working with parentheses. It allows us to multiply a single term across terms inside a parenthesis.
For instance, if you're given an expression like \((a + b) * c\), the distributive property lets us rewrite it as \(a*c + b*c\). This means you multiply each term inside the parenthesis by the term outside.
In the context of our exercise:

• The equation starts as \(x - 2.5 = 0.3(x - 2)\).
• We apply the distributive property: \(0.3 * x - 0.3 * 2\), which simplifies to \(0.3x - 0.6\).

By this method, you're able to transform equations and prepare them for further simplification. The distributive property is like a tool that unpacks expressions, making them easier to handle.

Combining Like Terms

Once we've used the distributive property to simplify a problem, the next step is combining like terms. This means identifying and merging terms that have the same variable or no variable at all.
In our example, after distributing, we end up with the equation \(x - 2.5 = 0.3x - 0.6\). Let's look at what combining like terms entails:

• You have terms like \(x\) and \(0.3x\). These can be combined on one side of the equation.
• You also have constants (numbers without variables), which should be combined on the opposite side.

So, by subtracting \(0.3x\) from both sides, you get \(0.7x - 2.5 = -0.6\). This step boils your equation down to its simplest form, readying it for solving the variable.
When you effectively combine like terms, you streamline the equation, ensuring each step you take is clear and purposeful.

Isolating Variables

Isolating the variable, such as 'x' in our scenario, refers to getting 'x' alone on one side of the equation. This is the heart of solving equations, as it gives the solution directly.
From the combined form \(0.7x - 2.5 = -0.6\), we aim to isolate \(x\):

• First, remove the constant next to \(x\) by performing the opposite operation. Here, that means adding \(2.5\) to both sides, leading to \(0.7x = 1.9\).
• Finally, to isolate \(x\), you divide both sides by \(0.7\), resulting in \(x = \frac{1.9}{0.7}\), which computes approximately to \(x \approx 2.714\).

Isolating the variable is like peeling away the layers around 'x' until it stands alone. Success at this step means you've reached the solution.