Problem 31 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation algebraically. Then write the equation in the form \(f(x)=0\) and use a graphing utility to verify the algebraic solution. $$\frac{3 x}{2}+\frac{1}{4}(x+2)=10$$

Step-by-Step Solution

Answer

The solution for the equation \(\frac{3 x}{2}+\frac{1}{4}(x+2)=10\) is \(x=5.4\).

Step 1: Solve the Equation Algebraically

Before solving, simplify the equation: \(\frac{3 x}{2}+\frac{1}{4}(x+2)=10\). Distribute \(\frac{1}{4}\) to both \(x\) and \(2\) in the bracket. Which gives us, \(\frac{3 x}{2}+\frac{1}{4}x+\frac{1}{2}=10\). Combine similar terms which are \(\frac{3 x}{2}\) and \(\frac{1}{4}x\). Our equation becomes \(\frac{7 x}{4}+\frac{1}{2}=10\). Subtract \(\frac{1}{2}\) from both sides. The simplified equation is \(\frac{7 x}{4}=9.5\).

Step 2: Solve for x

After simplification, Multiplying both sides of the equation by \(\frac{4}{7}\), yields \(x=5.4\).

Step 3: Rewrite the Equation in Form \(f(x)=0\)

Substitute \(x = 5.4\) in the equation and rewrite it in the form of \(f(x) = 0\). The rewritten equation should be \(f(x) = \frac{7}{4}x - 9.5\).

Step 4: Verify the Solution Using a Graphing Calculator

By inputting \(f(x) = \frac{7}{4}x - 9.5\) into a graphing utility, it can be confirmed that \(f(x) = 0\) when \(x = 5.4\).

Key Concepts

Algebraic Manipulation

When it comes to solving linear equations, algebraic manipulation is an essential skill. It involves rearranging the equation to isolate the variable and solve for it. Start with simplifying the equation by distributing any coefficients within parentheses and combining like terms. For example, given the equation \(\frac{3 x}{2}+\frac{1}{4}(x+2)=10\), distribute \(\frac{1}{4}\) to both \(x\) and \(2\), resulting in \(\frac{3 x}{2}+\frac{1}{4}x+\frac{1}{2}=10\).

• Combine similar terms, such as \(\frac{3 x}{2}\) and \(\frac{1}{4}x\), to get \(\frac{7 x}{4}+\frac{1}{2}=10\).
• Subtract \(\frac{1}{2}\) from both sides to isolate the term with the variable.
• Multiply or divide to get the variable alone, \(x\).

After these steps, we obtain \(\frac{7 x}{4}=9.5\), and by applying inverse operations, specifically multiplying both sides by \(\frac{4}{7}\), we find that \(x=5.4\). The power of algebraic manipulation lies in consistently applying basic arithmetic rules to both sides of the equation until the variable is isolated.

Graphing Calculator Verification

After solving an equation algebraically, a graphing calculator can be an excellent tool for verification. To check the algebraic solution of \(x=5.4\) for the given equation, we need to enter the equivalent function \(f(x)=\frac{7}{4}x - 9.5\) into the graphing calculator.

Step-by-Step Verification Process

• Start by turning on your graphing calculator and accessing the function input screen.
• Enter the function derived from our equation, \(f(x)=\frac{7}{4}x - 9.5\).
• Use graphing tools to locate the x-intercept, which is the point where \(f(x)=0\).

The x-intercept should be visible on the graph as the point where the line crosses the x-axis. If the graph shows the intercept at \(x=5.4\), it corroborates our previous algebraic solution, proving the effectiveness of the algebraic manipulation.

Linear Functions

A linear function is one of the most fundamental concepts in algebra, characterized by a constant rate of change, which graphically represents a straight line. Every linear equation can be written in the form of a function, usually expressed as \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.In the context of our problem, we can write the simplified equation \(\frac{7 x}{4}=9.5\) in the function form as \(f(x) = \frac{7}{4}x - 9.5\), ready for graphing and analysis. The slope \(\frac{7}{4}\) indicates how steeply the line rises or falls, while \( -9.5\) determines where it crosses the y-axis.The utility of understanding linear functions extends beyond just solving equations. It allows students to predict and compare relationships between variables, perform trend analysis, and deeply understand the concept of rate and intercepts in various real-world scenarios.