Problem 75 - College Algebra with Modeling and Visualization

Question

Exercises \(75-82:\) Solve the equation (to the nearest tenth) (a) symbolically, (b) graphically, and (c) numerically. $$ 5-(x+1)=3 $$

Step-by-Step Solution

Answer

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

Step 1: Simplify the Equation

Start with the equation: \(5 - (x + 1) = 3\). Remove the parentheses by distributing the negative sign: \(5 - x - 1 = 3\). Simplify the equation by combining like terms: \(4 - x = 3\).

Step 2: Isolate the Variable

To solve for \(x\), subtract 4 from both sides of the equation: \(-x = 3 - 4\). This simplifies to: \(-x = -1\).

Step 3: Solve for x

Divide by \(-1\) to solve for \(x\): \(x = 1\).

Step 4: Graphically Solve the Equation

Graph the equations \(y = 5 - (x + 1)\) and \(y = 3\). The solution \(x = 1\) can be found where the two graphs intersect.

Step 5: Numerically Verify the Solution

Substitute \(x = 1\) back into the original equation: \(5 - (1 + 1) = 3\). Calculate: \(5 - 2 = 3\), confirming that both sides are equal. Therefore, \(x = 1\) is correct to the nearest tenth.

Key Concepts

Equation Solving

Solving equations is a fundamental concept in algebra. It involves finding the value of the variable that makes the equation true. Let's consider the equation given: \[ 5 - (x+1) = 3 \].The first step in solving an equation like this is to simplify it. We do this by removing the parentheses and distributing the negative sign over the terms inside the parentheses:

• Original equation: \( 5 - (x+1) \)
• Simplify to: \( 5 - x - 1 \)

Next, combine like terms. Here, we combine the constants on the left side of the equation:

• Simplified further to: \( 4 - x = 3 \)

To isolate the variable \(x\), manipulate the equation by performing the same operation on both sides. Subtract 4 from both sides to keep the equation balanced:

• \( -x = 3 - 4 \)
• \( -x = -1 \)

Finally, solve for \(x\) by dividing each side by \(-1\), yielding:

• \( x = 1 \).

This process demonstrates how to methodically solve for the unknown in simple linear equations.

Graphical Method

The graphical method for solving equations involves representing each side of the equation as a separate function and plotting these on a graph. The solution is where the graphs intersect.

For our example, plot the functions:

• \( y = 5 - (x+1) \)
• \( y = 3 \)

Plotting these functions step-by-step can also provide insight into solving more complex equations.

• The first function simplifies to \( y = 4 - x \), a straightforward linear function with a slope of \(-1\) and y-intercept of 4.
• The second function is the constant line \( y = 3 \).

When these two lines are drawn on the same set of axes, the intersection point indicates the value of \(x\) that solves the equation:

• The lines intersect at \( x = 1 \).

Seeing the solution visually can enhance understanding and offer a backup method for verification. By confirming that the lines cross at \( x = 1 \), we verify the solution derived algebraically.

Numerical Verification

Numerical verification is the process of plugging the solution back into the original equation to ensure accuracy. It is a crucial step to confirm the solution's correctness.For our equation:\[ 5 - (x+1) = 3 \], substituting \( x = 1 \) back into the equation gives:

• \( 5 - (1 + 1) = 3 \)
• \( 5 - 2 = 3 \)
• \( 3 = 3 \)

Both sides of the equation equate, confirming that \( x = 1 \) is indeed the correct solution. Numerical verification not only assures us of the solution's accuracy but also builds confidence in the algebraic methods used. Never skip this step to avoid possible errors or miscalculations.