Problem 53 - College Algebra with Modeling and Visualization

Question

Exercises \(53-58\) : Use the intersection-of-graphs method to solve the equation. Then solve symbolically.) x + 4 = 1 - 2x

Step-by-Step Solution

Answer

The solution of the equation is \(x = -1\).

Step 1: Interpret the Given Equation

The given equation is \(x + 4 = 1 - 2x\). To use the intersection-of-graphs method, we first need to identify the two sides of the equation as separate functions. Let \(f(x) = x + 4\) and \(g(x) = 1 - 2x\). Our goal is to find the value of \(x\) at which these two functions intersect.

Step 2: Graph the Functions

Graph the functions \(f(x) = x + 4\) and \(g(x) = 1 - 2x\) on a coordinate plane. The graph of \(f(x) = x + 4\) is a straight line with a slope of 1 and a y-intercept at 4. The graph of \(g(x) = 1 - 2x\) is a straight line with a slope of -2 and a y-intercept at 1.

Step 3: Determine the Intersection Point

By observing the graphs of \(f(x)\) and \(g(x)\), find the point where the two lines intersect. This intersection point will be where the two functions have the same value, indicating the solution to the equation.

Step 4: Solve Symbolically

To verify the solution found graphically, solve the equation symbolically: \[x + 4 = 1 - 2x\]Add \(2x\) to both sides:\[3x + 4 = 1\]Subtract 4 from both sides:\[3x = -3\]Divide by 3:\[x = -1\]

Step 5: Verify the Solution

Check the solution by substituting \(x = -1\) back into both functions: For \(f(x)\): \(-1 + 4 = 3\)For \(g(x)\): \(1 - 2(-1) = 3\)Since both functions yield the same value, the solution \(x = -1\) is confirmed.

Key Concepts

Intersection of Graphs

The intersection of graphs is a visual method used to solve equations by identifying where two function graphs meet. When we graph each side of an equation as a function, their intersection point gives us the solution. Using graphing, we convert the problem into finding where these lines cross each other on a coordinate plane.

In our example, we used two functions:

• \( f(x) = x + 4 \)
• \( g(x) = 1 - 2x \)

These linear functions each produce a straight line on the graph. The intersection occurs at the point where both lines have the same x-value and y-value simultaneously. This x-coordinate of the intersection is the solution to the equation. Graphical visualization not only provides a method to find solutions, but also allows us to better understand the nature of relationships defined by each linear equation.

By interpreting our graphs, we can visually see how changes in one variable could impact the other, offering a complementary understanding alongside purely symbolic solutions. This aids in developing a more intuitive grasp of the behavior of linear equations.

Linear Equations

Understanding linear equations is key in algebra as they are equations that make a straight line when graphed. A linear equation can generally be expressed in the form \( y = mx + b \), where:

• \( m \) is the slope of the line, dictating how steep the line is
• \( b \) is the y-intercept, showing where the line crosses the y-axis

In our example, \( f(x) = x + 4 \) has a slope \( m = 1 \) with a y-intercept at 4. This means for every increase of 1 in \( x \), \( f(x) \) increases by 1. Conversely, \( g(x) = 1 - 2x \) has a slope \( m = -2 \), indicating that \( g(x) \) decreases by 2 for every increase of 1 in \( x \), and a y-intercept at 1.

Understanding the properties of these equations helps in predicting their graph shapes and determining how their slopes and intercepts influence their orientation and position on the plane. Recognizing these patterns makes it easier to solve and analyze complex algebraic problems and equations.

Symbolic Manipulation

Symbolic manipulation is the process of rearranging and simplifying algebraic equations to find their solutions. This method requires a good grasp of algebraic principles and rules, allowing us to manipulate the equation's symbols to isolate the variable of interest.

Consider this equation:

• \( x + 4 = 1 - 2x \)

To solve it symbolically, we first move all \( x \)-terms to one side:

• Add \( 2x \) to both sides to obtain \( 3x + 4 = 1 \).

Next, get all constant terms on the other side by subtracting 4:

• This leaves \( 3x = -3 \).

Finally, we divide by 3 to solve for \( x \):

• \( x = -1 \).

After solving symbolically, it is prudent to verify by plugging this solution back into the original equation to ensure both sides equal. This solidifies our understanding and reassures us that the variable handled symbolically is indeed the correct solution.