Problem 25 - Algebra 1

Question

Solve the equation algebraically. Check your solution graphically. $$\frac{1}{2} x+5=3$$

Step-by-Step Solution

Answer

The algebraic solution to the equation is \(x = -4\). This is verified by the intersection of the two lines on the graph.

Step 1: Simplify the equation

The first step is to simplify the equation and isolate the variable x on one side. This can be done by subtracting 5 from both sides of the equation. \[ \frac{1}{2}x + 5 - 5 = 3 - 5 \] The equation simplifies to \(\frac{1}{2}x = -2 \).

Step 2: Solve for x

Next, to solve for x, multiply both sides of the equation by 2. This will cancel out the 1/2 on the left side, leaving x on its own. \[ 2 \times \frac{1}{2} x = 2 \times -2 \] As a result, we get \(x = -4 \).

Step 3: Graphical verification

In order to verify the solution graphically, we need to graph the original equation. On a graph paper, plot the line \(y=\frac{1}{2} x + 5 \) and a horizontal line \(y = 3 \). The point where these two lines intersect is where \(x = -4 \), verifying our algebraic solution.

Key Concepts

Graphical Verification in Solving Linear Equations

Graphical verification is a handy method to confirm the solution of a linear equation. Once you solve an equation algebraically, you can graph it to verify your answer visually. For example, if we have solved the equation \(\frac{1}{2} x + 5 = 3\) and found that \(x = -4\), we can use a graph to check this solution.

To do this, plot the line \(y = \frac{1}{2} x + 5\) on a graph. This line represents all the possible solutions of the equation. You should also draw a horizontal line at \(y = 3\) because that is the other side of your equation. The x-coordinate of the point where these two lines intersect will give you the value of \(x\).

• Draw the graph for both equations.
• Mark where they intersect.

By locating the intersecting point, you confirm that your algebraic solution \(x = -4\) is correct. Graphing helps understand equations geometrically, providing a visual proof of your solution's accuracy.

Isolating Variables: Step-by-Step Approach

When solving linear equations, one of the key steps is isolating the variable. This simply means getting the unknown variable, often represented as \(x\), alone on one side of the equation. Here's how you can effectively isolate a variable:

Consider the equation \(\frac{1}{2} x + 5 = 3\). Your first goal is to remove the constant term from the side containing \(x\). You achieve this by subtracting 5 from both sides:

\[\frac{1}{2} x + 5 - 5 = 3 - 5\]
Simplifying this gives the result \(\frac{1}{2} x = -2\).

• Identify the numbers or terms that are not linked directly with the variable.
• Use addition or subtraction to move these terms to the opposite side.

By isolating \(x\), it becomes easier to handle and you are one step closer to finding its value. This process of isolation lays the groundwork for solving any further operations needed to unravel the equation.

Multiplying Equations for Simplification

Once you've isolated the variable in a linear equation, the next step usually involves ensuring that the variable stands alone, without any coefficients or fractions. That’s where multiplying equations comes in handy.

In our example, you have \(\frac{1}{2} x = -2\). To solve for \(x\), multiply both sides by 2 to cancel out the fraction. This action eliminates the coefficient, leaving you with:

\[2 \times \frac{1}{2} x = 2 \times -2\]
This simplifies to \(x = -4\).

• Identify the fraction or number multiplier next to \(x\).
• Choose the reciprocal or inverse operation to eliminate it.

Multiplying both sides by the same number keeps the equation balanced, a fundamental principle when solving equations. This technique is not only effective in clearing fractions but also in handling equations with multiple terms or coefficients, turning them into simple, solvable forms.