Problem 29 - Algebra Form and Function

Question

Does the equation have no solution, one solution, or an infinite number of solutions? $$ 4 x+3=7 $$

Step-by-Step Solution

Answer

Answer: The equation has one solution, x = 1.

Step 1: Isolate the variable x

To isolate x, we need to get it on one side of the equation. We can start by subtracting 3 from both sides: $$ 4x + 3 - 3 = 7 - 3 \\ 4x = 4 $$

Step 2: Solve for x

Now that we have x isolated, we can determine its value by dividing both sides of the equation by 4: $$ \frac{4x}{4} = \frac{4}{4} \\ x = 1 $$ Since we were able to find a value for x, the equation has one solution, x = 1.

Key Concepts

One Solution

When solving a linear equation, it's essential to determine the number of solutions it might have. A linear equation like \(4x + 3 = 7\) can have one solution, no solution, or infinitely many solutions. In this particular case, the equation has one solution.

This means there is exactly one value of \(x\) that makes the equation true. When you perform algebraic manipulations and end up with a specific value for \(x\), such as \(x = 1\), it indicates a single unique solution.

Recognizing this is key when analyzing the nature of linear equations.

Isolation of Variables

Isolation of variables is a crucial step in solving equations. It involves rearranging the equation to get the unknown variable by itself on one side. This makes it easier to see what value will satisfy the equation.

Consider the example: \(4x + 3 = 7\).

• Start by removing constants from one side. Here, subtract 3 from both sides:
• \(4x + 3 - 3 = 7 - 3\)
• This simplifies to \(4x = 4\).

Now the variable \(x\) is isolated with a coefficient of 4. This step sets the stage for finding the solution.

Algebraic Manipulation

Algebraic manipulation is the process of transforming an equation into a simpler form to find the solution. It's all about using mathematical operations strategically.

Once you've isolated the variable (as in \(4x = 4\)), the next step is to perform operations to solve for \(x\).

• Divide both sides by the coefficient (4 in this case):
• \(\frac{4x}{4} = \frac{4}{4}\)

This simplifies to \(x = 1\), providing a clear solution.

Understanding how to manipulate equations allows you to handle more complex algebraic problems efficiently.