Problem 56 - College Algebra with Modeling and Visualization

Question

Use the intersection-of-graphs method to solve the equation. Then solve symbolically. 1-2x=x+4

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Identify Expressions to Graph

First, we separate the equation into two expressions, one for each side of the equation: \( y_1 = 1-2x \) and \( y_2 = x+4 \). We'll graph these two expressions to find their intersection.

Step 2: Graph the Equations

Plot the two lines on a coordinate plane. The line for \( y_1 = 1-2x \) is a downward sloping line with a y-intercept at 1. The line for \( y_2 = x+4 \) is an upward sloping line with a y-intercept at 4.

Step 3: Find the Intersection Point

Determine where the two graphs intersect. This point represents the solution to the equation \( 1-2x = x+4 \). Visually from the graph, the intersection appears to be at \( x = -1 \).

Step 4: Solve Symbolically

Set the two expressions equal to each other to solve symbolically: \[ 1 - 2x = x + 4 \] Subtract \( x \) from both sides:\[ 1 - 3x = 4 \] Subtract 1 from both sides:\[ -3x = 3 \] Divide by -3:\[ x = -1 \]

Step 5: Verify Solution

Substitute \( x = -1 \) back into the original equation to verify: \( 1 - 2(-1) = (-1) + 4 \) becomes \( 1 + 2 = 3 \) and \( -1 + 4 = 3 \), which confirms the equation.

Key Concepts

Graphical Solution

When solving equations using the intersection of graphs method, we use visual aids by drawing graphs. This method makes abstract concepts more tangible.
To begin, we separate the original equation into two distinct parts. In our example, we have the equations:

• \( y_1 = 1 - 2x \)
• \( y_2 = x + 4 \)

We then plot these equations on the same coordinate plane. The goal is to identify where the graphs intersect. This intersection point corresponds to the solution for the variables in the original equation.
The equation \( y_1 = 1-2x \) creates a downward sloping line since its rate, or coefficient of \( x \), is negative. On the other hand, \( y_2 = x + 4 \) has an upward slope, indicating a positive relationship between \( x \) and \( y \). Finding the intersection visually gives us an intuitive understanding of where both equations hold true simultaneously.

Symbolic Solution

In contrast to graphical solutions, symbolic solutions involve algebraic manipulation without visual aids. This allows for precise solutions, provided calculations are done accurately.
To solve symbolically, set the expressions you graphed equal to each other because their intersection point in the graph is where they are equal. For our equations, this means:

• \( 1 - 2x = x + 4 \)

Next, simplify by manipulating the equation:

• Subtract \( x \) from both sides: \( 1 - 3x = 4 \)
• Subtract 1 from both sides: \( -3x = 3 \)
• Divide by -3: \( x = -1 \)

These steps yield the solution for \( x \) directly. The elegance of the symbolic method lies in its ability to provide exact and efficient solutions.

Equation Solving

Equation solving is about finding the value of \( x \) that balances both sides of the equation, ensuring each transformation maintains equality.
Transformations used during solving include:

• Addition or subtraction to eliminate constants or terms
• Multiplication or division to simplify the coefficient of \( x \)

Each step should make the equation simpler until \( x \) is isolated on one side.
For our example, subtracting \( x \) combines like terms, while subtracting constants streamlines the equation further. Dividing by the coefficient of \( x \), finally, isolates it, clearly stating \( x = -1 \) as the solution.
Verification by substituting \( x = -1 \) back into the original equation reassures us of the accuracy of our symbolic solution.

Coordinate Geometry

Coordinate geometry, or analytic geometry, uses a coordinate plane to understand and solve problems involving geometric figures and equations.
Key components are:

• Coordinates (x, y): They define points on the plane.
• Lines and curves: Represent equations visually.
• Intersection points: Solutions to equations represented by the graphs.

In our equation-solving context, the coordinate plane provides a way to depict the equations' behaviors. An intersection of two lines on this plane means their algebraic forms are equal at that point, therefore solving the equation.
Coordinate geometry supports understanding by linking algebraic equations with tangible, visual graphs, making abstract solutions more concrete and intuitive.