Problem 79 - College Algebra with Modeling and Visualization

Question

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ x-3=2 x+1 $$

Step-by-Step Solution

Answer

The solution is \(x = -4\) (to the nearest tenth).

Step 1: Rearrange the Equation

Start by rearranging the equation to get all terms involving \(x\) on one side of the equation. The original equation is \(x - 3 = 2x + 1\). Subtract \(2x\) from both sides to isolate the terms with \(x\) on the left-hand side: \(x - 2x - 3 = 1\), which simplifies to \(-x - 3 = 1\).

Step 2: Solve for x

Add 3 to both sides of the equation to isolate the term with \(x\): \(-x - 3 + 3 = 1 + 3\). This simplifies to \(-x = 4\). Multiply both sides by \(-1\) to solve for \(x\): \(x = -4\).

Step 3: Check the Solution Symbolically

Substitute \(x = -4\) back into the original equation to verify the solution. See if both sides equal when \(x = -4\): \((-4) - 3 = 2(-4) + 1\). Simplifying, both sides equal \(-7\). Thus, \(x = -4\) is the symbolic solution.

Step 4: Solve Graphically

Sketch or use graphing software to graph the functions \(y = x - 3\) and \(y = 2x + 1\). Find the x-coordinate where both graphs intersect. The intersection point is at \(x = -4\), confirming the solution.

Step 5: Solve Numerically

Evaluate numerically by substituting values starting from an estimated range around \(x = -4\) and refining the guess. Plug in \(-4\) and calculate both sides: Left side \((-4) - 3 = -7\) and Right side \(2(-4) + 1 = -7\). The values are equal, confirming \(x = -4\) is the solution.

Key Concepts

Symbolic Solution

Solving equations symbolically involves finding an exact solution through algebraic manipulation. In this problem, we need to rearrange the terms to isolate the variable, which allows us to solve for its exact value.

• Start by getting all terms with the variable on one side: begin with the equation \(x - 3 = 2x + 1\).
• Subtract \(2x\) from both sides to obtain \(-x - 3 = 1\).
• Next, add 3 to each side, simplifying to \(-x = 4\).
• Multiply by -1 to solve for \(x\), resulting in \(x = -4\).

Checking the solution by substituting \(x = -4\) back into the original equation confirms its correctness, as both sides equate to \(-7\). This method showcases algebraic techniques to gain the symbolic solution.

Graphical Solution

A graphical solution involves using a graph to find where two lines intersect, which represents the solution to the equation. This is particularly helpful for visual learners.

• To begin, graph both equations: \(y = x - 3\) and \(y = 2x + 1\).
• Use graphing software or sketch the lines on graph paper.
• The solution to the equation is where these lines intersect on the graph.

For our equation, the intersection occurs at \(x = -4\). This means both lines meet at that x-coordinate, confirming the value found symbolically. Graphical solutions offer a visual perspective that complements algebraic techniques.

Numerical Solution

The numerical solution approach involves estimating and calculating values to refine an answer. This method can be useful when exact solutions are difficult to find or when verifying answers.

• Start with an approximation; in this case, we suspect \(x\) is around \(-4\).
• Plug \(-4\) into the original equation to check: Left side \((-4) - 3 = -7\), Right side \(2(-4) + 1 = -7\).
• Both sides equal \(-7\), showing that \(x = -4\) is indeed correct.

Numerical solutions can involve more advanced techniques, like iteration or using computational tools, for greater accuracy. Here, it confirmed the symbolic and graphical solutions effectively.

Algebraic Techniques

Algebraic techniques involve manipulating equations to isolate and solve for variables. These are essential for finding symbolic solutions and can be applied to a variety of equation types.

• Begin with rearranging terms to bring variables to one side: \(x - 3 = 2x + 1\).
• Combine like terms by subtracting \(2x\) from both sides, leading to: \(-x - 3 = 1\).
• Isolate \(x\) by adding 3 to both sides and simplifying: \(-x = 4\).
• Finally, multiply by \(-1\) to find \(x = -4\).

These skills are foundational in algebra and provide the means to solve much more complex equations similarly. Through step-by-step manipulation, algebraic techniques bridge the gap between problem and solution.