Problem 20 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ 5(x+3)+9=-2(x-2)-1 $$

Step-by-Step Solution

Answer

The solution is \(x = -3\).

Step 1: Expand Both Sides

First, distribute the numbers outside the parentheses on both sides of the equation:For the left side, distribute 5:\[5(x+3) = 5x + 15\]For the right side, distribute -2:\[-2(x-2) = -2x + 4\] Substituting these into the equation gives us:\[5x + 15 + 9 = -2x + 4 - 1\]

Step 2: Simplify Both Sides

Combine like terms on each side:On the left side: \[5x + 24\](On the right side, \(4 - 1 = 3\).) The equation becomes:\[5x + 24 = -2x + 3\]

Step 3: Move Variables to One Side

Add \(2x\) to both sides to move all terms involving \(x\) to the left side:\[5x + 2x + 24 = 3\]This simplifies to:\[7x + 24 = 3\]

Step 4: Isolate the Variable

Subtract 24 from both sides to isolate terms with \(x\):\[7x = 3 - 24\]This simplifies to:\[7x = -21\]

Step 5: Solve for x

Divide both sides by 7 to solve for \(x\):\[x = \frac{-21}{7}\]This simplifies to:\[x = -3\]

Step 6: Conclusion

The final value of \(x\) is \(-3\). We have solved the equation for the variable.

Key Concepts

Distributive Property

The distributive property is a fundamental concept used when solving equations like the one in our exercise: \(5(x+3)+9=-2(x-2)-1\). This property allows us to remove parentheses by distributing a multiplier across terms inside the parentheses.

For example, on the left side of the equation, the term \(5(x+3)\) requires us to distribute the 5:

• Multiply 5 by \(x\): \(5 \cdot x = 5x\)
• Multiply 5 by 3: \(5 \cdot 3 = 15\)

Bringing these together, \(5(x+3)\) becomes \(5x + 15\).

Similarly, on the right side, distributing \(-2\) gives:

• \(-2 \cdot x = -2x\)
• \(-2 \cdot -2 = 4\)

Here, \(-2(x-2)\) simplifies to \(-2x + 4\). The distributive property simplifies complex expressions and sets the stage for further simplification.

Combining Like Terms

After using the distributive property, equations often have terms that can be combined to make the equation simpler. This process is known as combining like terms.

Let's look at the updated equation: \(5x + 15 + 9 = -2x + 4 - 1\). Combining like terms involves adding or subtracting numbers and variables that are similar.

• On the left side, combine the constants: \(15 + 9 = 24\), leading to \(5x + 24\).
• On the right side, simplify the constants: \(4 - 1 = 3\), resulting in \(-2x + 3\).

Now, our equation is simplified to \(5x + 24 = -2x + 3\). Combining like terms reduces clutter and reveals a clearer path to solve for the variable.

Isolating the Variable

With the equation \(5x + 24 = -2x + 3\), the next step is to isolate the variable. This means getting all \(x\) terms on one side and constants on the other.

Start by moving the \(-2x\) term to the left side:

• Add \(2x\) to both sides: \(5x + 2x + 24\).
• The equation becomes \(7x + 24 = 3\).

Next, move the constant 24 to the right side:

• Subtract 24 from both sides: \(7x = 3 - 24\).
• Simplify right side: \(7x = -21\).

These steps ensure that \(x\) is by itself on one side, making it easier to solve for its value.

Equivalent Equations

In the process of solving equations, we often create equivalent equations. These are equations which, despite looking different, have the same solution. By performing operations such as distributing, combining like terms, and isolating the variable, we ensure the equation stays equivalent.

For each step, we maintain equivalency by:

• Preserving the balance of the equation through operations (adding, subtracting, multiplying, etc.).
• Ensuring any operation performed on one side is also done on the other side.

These operations keep transformations such as \(5(x+3) = 5x + 15\) and \(7x = -21\) equivalent to the original equation.

The final simplified equation, \(x = -3\), holds the same truth as our initial one, affirming the consistency in problem-solving.