Problem 76 - College Algebra with Modeling and Visualization

Question

Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 7-(3-2 x)=1 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -1.5\).

Step 1: Simplify the Equation

Firstly, let's simplify the left side of the equation. Start by distributing the negative sign inside the parenthesis: \[ 7 - 3 + 2x = 1 \] This simplifies to: \[ 4 + 2x = 1 \]

Step 2: Isolate the Variable

To isolate the variable \(x\), subtract 4 from both sides of the equation: \[ 2x = 1 - 4 \] Simplify the right side: \[ 2x = -3 \] Next, divide both sides by 2: \[ x = -\frac{3}{2} \] This results in: \[ x = -1.5 \]

Step 3: Graphical Solution Approach

To solve graphically, plot the equations \(y_1 = 7 - (3 - 2x)\) and \(y_2 = 1\) on the same set of axes. The x-coordinate of the intersection point of these two lines will give the solution to the equation. After graphing, you'll find the intersection at \(x = -1.5\).

Step 4: Numerical Verification

Substitute \(x = -1.5\) back into the simplified equation \(4 + 2x = 1\) to check:\[ 4 + 2(-1.5) = 1 \] This simplifies to: \[ 4 - 3 = 1 \] This holds true, so \(x = -1.5\) is the correct solution.

Key Concepts

Symbolic Solution

In algebra, symbolic solutions refer to solving equations through manipulation and simplification using algebraic symbols and operations. The first step when tackling an equation like \(7-(3-2x)=1\) is to simplify it. We rewrite the equation by distributing the negative sign across the terms within the parentheses, leading to the expression \(7 - 3 + 2x = 1\). This simplifies to \(4 + 2x = 1\).

Once simpified, we focus on isolating the variable. Subtract 4 from each side to obtain \(2x = 1 - 4\), which gives us \(2x = -3\). Finally, dividing both sides by 2 isolates \(x\), resulting in \(x = -1.5\). In symbolic solutions, each step logically follows from the previous, using rules of arithmetic and algebra to uncover the variable's value. This approach is methodical and exact, leading to a precise solution.

Graphical Solution

Graphical solutions involve visualizing equations on a graph to find their intersection points. For the equation \(7-(3-2x)=1\), consider two functions: \(y_1 = 7 - (3 - 2x)\) and \(y_2 = 1\). The solution to the equation corresponds to the x-value at which these two functions intersect on a graph.

When plotting these functions, the graphical intersection is seen at \(x = -1.5\). This visual method confirms the symbolic solution, as the x-coordinate of the intersection yields the equation's solution. The graphical approach provides a geometric confirmation of the solution, offering a clearer understanding of the relationship between variables. It is especially helpful when dealing with complex or multiple equations, as it can visually display potential solutions.

Numerical Solution

Numerical solutions use methods like substitution or iterations to validate a proposed solution. Once we have the symbolic solution \(x = -1.5\), we substitute it back into the original simplified equation \(4 + 2x = 1\) for confirmation. Plugging in \(x = -1.5\), we calculate \(4 + 2(-1.5)\) which simplifies to \(4 - 3 = 1\). This checks out, indicating the solution is correct.

In cases where exact solutions are difficult to determine algebraically or graphically, numerical methods can offer approximations. They involve estimating values through iterations or using technology like calculators and computers to handle lengthy calculations. Numerical solutions can be essential for equations involving complex numbers or non-linear systems, providing a practical way to find and verify solutions.