Problem 80 - College Algebra with Modeling and Visualization

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \( x = 2 \).

Step 1: Expand the Equation

The given equation is \( 3(x-1) = 2x - 1 \). First, we'll expand the left-hand side: \[ 3(x-1) = 3x - 3 \] So, the equation becomes: \[ 3x - 3 = 2x - 1 \]

Step 2: Rearrange the Equation

Subtract \( 2x \) from both sides to move all terms involving \( x \) to one side: \[ 3x - 2x - 3 = -1 \] This simplifies to: \[ x - 3 = -1 \]

Step 3: Solve for x

Add 3 to both sides to isolate \( x \): \[ x - 3 + 3 = -1 + 3 \] Simplifying gives: \[ x = 2 \] Therefore, the solution to the equation is \( x = 2 \).

Step 4: Graphical Solution

Graph the equations \( y = 3(x-1) \) and \( y = 2x - 1 \) and find the x-coordinate of their intersection. This point will confirm the solution. Both lines intersect at the point \( (2, 3) \), confirming that \( x = 2 \).

Step 5: Numerical Verification

Substitute \( x = 2 \) back into the original equation to verify:Left side calculation: \( 3(2-1) = 3 \times 1 = 3 \)Right side calculation: \( 2 \times 2 - 1 = 4 - 1 = 3 \)Since both sides are equal, the solution \( x = 2 \) is correct.

Key Concepts

Symbolic Solving

Symbolic solving involves manipulating equations algebraically to find the exact value of the unknown variable. It's almost like solving a puzzle with letters and numbers! Let's break it down further using our equation given by problem: Start with the original equation: \[3(x - 1) = 2x - 1\]1. **Expand the Equation:** - We expand the left side: \( 3(x - 1) = 3x - 3 \). - So, it becomes: \( 3x - 3 = 2x - 1 \).2. **Rearrange Terms:** - Subtract \( 2x \) from each side to collect all \( x \)s together: \( 3x - 2x - 3 = -1 \). - Simplify to get: \( x - 3 = -1 \).3. **Isolate the Variable:** - Add 3 to both sides so that \( x \) stands alone: \( x = 2 \).And there you go, solving symbolically! Symbolic methods give us clarity and precision without needing to refer to technology or drawings. They're like the algebraic backbone of the equation world.

Graphical Methods

Graphical methods provide us a visual way of understanding equations by turning them into pictures. It's like drawing a map, where different lines and curves meet. Here is how we could look at our equation graphically:First, we need to break the equation into two parts, similar to what we see in a puzzle. These parts are functions of \( y \): - \( y = 3(x - 1) \) for the left side.- \( y = 2x - 1 \) for the right side.**Plotting the Equations:**- Draw a graph with an x-axis and a y-axis. For both equations, plug in various values of \( x \) to find corresponding \( y \) values. - Plotting these values will give you two straight lines if done accurately.**Finding the Intersection:**- Look for the point where the two lines meet. This x-coordinate of the intersection represents the solution of the equation.- In our case, these lines intersect at \( x = 2 \), confirming the symbolic solution we found earlier.Graphing is helpful when you're a visual learner or want to verify a solution quickly without doing the algebra. Remember, it takes practice to become skilled at reading these visual cues!

Numerical Verification

Numerical verification acts as our final check—like double-checking your homework for mistakes. It's not just about trust; it's about certainty and confidence. Let's see how we can verify by plugging it back in:We've already solved for \( x = 2 \). Now, we substitute this back into the original equation to see if it's true:1. **Check the Left Side:** - Compute with \( x = 2 \): \( 3(2 - 1) = 3 \times 1 = 3 \).2. **Check the Right Side:** - Plug \( x = 2 \): \( 2 \times 2 - 1 = 4 - 1 = 3 \).Both sides should produce the same number if our initial solution is correct. And they do just that—both equal 3. This match indicates we've found the right solution. Numerical verification is crucial as it helps identify errors or missteps in solving. It's a comfort blanket that reassures us our symbolic work and graphical interpretation are spot on. Don’t skip it, because assurance is key!