Problem 37 - Algebra 1

Question

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

Step-by-Step Solution

Answer

The solution of the equation \(\frac{1}{2} x+5=3\) is \(x=-4\)

Step 1: Draw Graph of Function

Represent the equation by drawing the line \(\frac{1}{2} x+5=3\) on the cartesian plane. A common way to do this is to find points on the line, for instance, by assuming certain x-values and computing the corresponding y-values or vice versa. The line intersects the x-axis when y=0.

Step 2: Determine x-intercept

To solve the equation graphically, locate the x-intercept of the graph. This is the point where the graph intersects the x-axis. The x-coordinate of this point will be the solution to your equation.

Step 3: Check algebraically

First, subtract 5 from both sides of the equation: \(\frac{1}{2} x=3-5=-2\). Then, multiply every term by 2 to isolate x: \(x = -2*2 = -4\). This is the solution obtained algebraically. Verify if it coincides with the solution obtained graphically by plugging this value back into the given equation.

Key Concepts

Understanding the x-intercept

The x-intercept is an essential concept when solving equations graphically. It is the point on a graph where the line crosses the x-axis. At this point, the value of y is 0, which means you're solving for x when the output of the equation is zero. To identify the x-intercept from a graph, follow these steps:

• Draw the line representing your equation on the graph.
• Look for the point where this line intersects the x-axis.
• The x-coordinate of this intersection is the x-intercept.

Finding the x-intercept is often a straightforward method to visually determine the solution of a linear equation without doing extensive calculation, especially if the graph is accurately drawn.

Navigating the Cartesian Plane

A cartesian plane is a two-dimensional graphing system made up of two axes: the horizontal x-axis and the vertical y-axis. Each axis represents a number line, allowing you to graph equations and locate points. Here’s how you use it to solve equations graphically:

• The x-axis runs left to right and contains all possible values of x.
• The y-axis runs up and down and contains all possible values of y.
• You plot points by moving along the x-axis first, then parallel to the y-axis.

The origin, where these axes intersect, is the point (0, 0). Using a cartesian plane, you can visually represent equations by plotting lines or curves based on the values of x and y, such as the line from our example: \ \( \frac{1}{2}x + 5 = 3 \). A graph makes it easier to see where lines meet the axes or each other.

Grasping Linear Equations

Linear equations are mathematical expressions that form straight lines when graphed on a cartesian plane. They follow the general format of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. These equations are characterized by:

• A constant slope, \( m \), which indicates the line's steepness.
• The y-intercept, \( b \), which is the value where the line crosses the y-axis.
• A direct definition of a straight path on the graph.

In our example, the equation is re-arranged to: \( y = \frac{1}{2}x + 5 \). By graphing this line, the x-intercept can be easily located, simplifying the solution finding process. Linear equations are some of the most fundamental concepts in algebra, frequently serving as stepping stones for more complex calculations.

Performing Algebraic Verification

Algebraic verification is a crucial step after solving equations graphically. It ensures the graphical solution is accurate through mathematical calculations. For the given equation \( \frac{1}{2}x + 5 = 3 \), here's how the algebraic checking works:

• First, isolate x by subtracting 5 from both sides: \( \frac{1}{2}x = 3 - 5 = -2 \).
• Next, remove the fraction by multiplying all terms by 2: \( x = -2 \cdot 2 = -4 \).

By substituting \( x = -4 \) back into the original equation, validate the left side equals the right side. This verification provides confidence in your graphical solution, confirming consistency between different solving techniques. Algebraic verification acts as a dependable safety net for ensuring that your graphical observations are correct.