Problem 41 - Algebra 1

Question

Use a graphing calculator to find the solution of the equation. Check your solution algebraically. $$0.7(3 x-20)=22-3.9 x$$

Step-by-Step Solution

Answer

The correct solution would be given by the x-coordinate of the point where the graph crosses the X-axis. The specific value needs to be calculated using a graphing calculator and then confirmed algebraically by plugging it into the original equation.

Step 1: Rearrange the equation

First, rearrange the equation to have both sides equal to zero. So, the given equation \(0.7(3x - 20) = 22 - 3.9x\) becomes \(0.7(3x - 20) + 3.9x - 22 = 0\).

Step 2: Graph the equation

Next, plot the equation \(y = 0.7(3x - 20) + 3.9x - 22\) on the Y-axis of the graphing calculator. This gives you a visual representation of the equation.

Step 3: Find the intersection point

Look for the point where the graph intersects the X-axis. This intersection represents the solution to the equation, as it is the value of x that makes the equation equal to zero.

Step 4: Check solution algebraically

Finally, substitute the found value of x in the main equation to verify that it is indeed the solution. If the left-hand side equals the right side when the solution is substituted, then it is the correct answer.

Key Concepts

Algebraic Solution

An algebraic solution involves manipulating an equation using algebraic techniques to find the value of the unknown variable. In the given problem, the equation is presented in the form of \( 0.7(3x - 20) = 22 - 3.9x \). To solve this algebraically, you first rearrange the terms to one side of the equation, making it: \( 0.7(3x - 20) + 3.9x - 22 = 0 \).
Following this rearrangement, you can simplify further by distributing and combining like terms. Notice how these steps are crucial for setting up the mathematical groundwork, which will eventually let you solve for \(x\). The algebraic solution validates the results obtained by other methods, such as using a graphing calculator, ensuring that the solution is accurate and verified.

Linear Equations

Linear equations are fundamental in algebra involving variables raised to the power of one. This means that the graph of a linear equation is a straight line when plotted on a graph. In our exercise, the equation \(0.7(3x - 20) + 3.9x - 22 = 0\) is a linear equation.

• Each component of the equation maintains a constant rate of change, which is graphically represented as a line.
• Linear equations are structured in a standard form \( Ax + By = C \), but can appear in various forms by rearranging the terms.

Understanding the properties of linear equations helps in quick identification and solving similar problems using both graphing and algebraic methods.

Intersection Point

The intersection point in the context of solving equations graphically is where the graph of the equation crosses the X-axis. This occurrence indicates that the value of \( y \) in the equation is zero.
For the given exercise, plotting the linear equation \( y = 0.7(3x - 20) + 3.9x - 22 \) on a graphing calculator shows where it intersects the X-axis, which corresponds to the solution for \( x \).

• The intersection point represents the root or solution of the equation.
• Finding the intersection point graphically provides a visual confirmation of where the solution lies within the domain.

This graphical approach not only validates the algebraic result but also helps in visualizing the process for learners.

Solving Equations

Solving equations involves finding the values of variables that satisfy the equations, meaning that they make the left and right sides of the equality sign equal. For our exercise, once the equation is rearranged to \( 0.7(3x - 20) + 3.9x - 22 = 0 \), solving it involves either graphically finding the intersection point or algebraically manipulating it.

• Algebraically, you simplify terms sequentially, isolate the variable \( x \), and calculate its precise value.
• Graphically, with a graphing calculator, you're looking for where the graph touches or crosses the X-axis.

Each method offers an insight into the structure and nature of the equation, with graphic solutions offering an intuitive comprehension and algebraic solutions affirming precision.