Problem 38 - Algebra 1

Question

Use a graphing calculator to find the solution of the equation. Check your solution algebraically. $$4(x+2)=3(x+5)$$

Step-by-Step Solution

Answer

The solution to the equation \(4(x+2) = 3(x+5)\) is x = 7.

Step 1: Simplify the equation

Expand the parentheses to simplify your expression: \(4x+8 = 3x + 15\).

Step 2: Isolate the variable X

Subtract 3x from both sides to isolate the variable term on one side and constants on the other side: \(4x - 3x = 15 - 8\), resulting in \(x = 7\).

Step 3: Check the Solution

Substitute x = 7 back into the original equation: \(4(7+2) = 3(7+5)\), which simplifies to \(36 = 36\). Since both sides are equal, the solution is correct.

Key Concepts

Using a Graphing Calculator

A graphing calculator can be a very helpful tool when solving linear equations like \( 4(x+2)=3(x+5) \). It provides a visual representation of the equation and allows for an easier understanding of how the solution is derived.
To use a graphing calculator for this purpose, you would first rearrange the equation into two separate functions: \( y_1 = 4(x+2) \) and \( y_2 = 3(x+5) \). Then, by plotting these functions on the calculator's graph, you can find the point where the two lines intersect.
This point of intersection represents the solution to the equation, as it illustrates the value of \( x \) where both expressions are equal.

• Enter each side of the equation as separate equations in the calculator.
• Graph both equations within a suitable range to clearly see where they cross.
• Use the 'trace' or 'intersection' feature to pinpoint the exact value of \( x \) at their intersection.

Through this visual technique, a graphing calculator offers not only an answer but also a greater understanding of how the solution applies within a coordinate system.

Algebraic Solution

Solving linear equations algebraically involves manipulating the equation to isolate the variable. Let's break it down using the example \( 4(x+2)=3(x+5) \).
The first step is to expand the parentheses:

• Left Side: \( 4(x+2) \) becomes \( 4x + 8 \)
• Right Side: \( 3(x+5) \) becomes \( 3x + 15 \)

Next, you want to collect all terms involving the variable on one side and constant terms on the other. Achieve this by subtracting \( 3x \) from both sides:
\( 4x + 8 - 3x = 3x + 15 - 3x \) simplifies to \( x + 8 = 15 \).
Lastly, further simplify by subtracting \( 8 \) from both sides to solve for \( x \):
\( x = 15 - 8 \). So, \( x = 7 \).
This step-by-step process showcases the power of algebraic manipulation to arrive at the correct solution, highlighting precision and validation by substituting back to verify.

Simplifying Expressions

Simplifying expressions is a key step in solving equations, making them easier to handle. When we look at the equation \( 4(x+2)=3(x+5) \), simplification begins with expanding the terms inside the parentheses.
Expansion involves distributing the coefficient across each term inside the parentheses. For instance, \( 4(x+2) \) expands to \( 4x + 8 \) and \( 3(x+5) \) becomes \( 3x + 15 \).
By performing this, each side of the equation is broken down into more manageable parts, resulting in a clearer path to isolate the variable.
Follow these steps:

• Distribute the multipliers (coefficients) to each term inside the parentheses.
• Combine like terms on each side to simplify further.
• Proceed to isolate the variable if necessary.

Simplifying expressions is vital as it reduces complexity and allows easier elimination of terms to solve for the unknown. It's a tactical approach ensuring that each transformation keeps the equation equivalent.