Problem 62 - Elementary Algebra

Question

Solve. $$x+2 x 2-5 x+4+x+2 x 2+x-2=x-1 x 2-2 x-8$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{10}{9}\).

Step 1: Expand the Equation

First, expand the equation. Note that any multiplication in the form \( x2 \) should be interpreted as two times \( x \). Convert and rewrite as:\[(x + 2(x \cdot 2) - 5x + 4 + x + 2(x \cdot 2) + x - 2 = x - (x \cdot 2) - 2x - 8) \]Simplifying the multiplications, we get:\[x + 4x - 5x + 4 + x + 4x + x - 2 = x - 2x - 2x - 8\]

Step 2: Combine Like Terms on both Sides

Next, combine like terms on both sides of the equation.On the left side:\[(1x + 4x - 5x + 1x + 4x + 1x) + (4 - 2) = (1 + 4 - 5 + 1 + 4 + 1)x + 2 = 6x + 2\]On the right side:\[(1x - 2x - 2x) + (-8)= (1 - 2 - 2)x - 8 = -3x - 8\]

Step 3: Write the Simplified Equation

Now write the simplified form of the equation:\[6x + 2 = -3x - 8\]

Step 4: Add 3x to Both Sides

Add \(3x\) to both sides of the equation to start isolating x on one side:\[6x + 3x + 2 = -3x + 3x - 8 \]This simplifies to:\[9x + 2 = -8\]

Step 5: Subtract 2 from Both Sides

Subtract 2 from both sides to further isolate terms involving x:\[9x + 2 - 2 = -8 - 2\]This simplifies to:\[9x = -10\]

Step 6: Solve for x

Divide both sides by 9 to solve for \(x\):\[\frac{9x}{9} = \frac{-10}{9}\]Hence, the solution is:\[x = -\frac{10}{9}\]

Step 7: Verify the Solution

Substitute \(x = -\frac{10}{9}\) back into the original equation to ensure that both sides are equal. Simplifying both sides should yield an equivalent value, confirming the answer is correct.

Key Concepts

Combining Like Terms

When solving algebraic equations, one essential step is combining like terms. Like terms are terms that have the same variable raised to the same power. For example, in the equation \(x + 4x - 5x + 4 + x + 4x + x - 2\), you can combine all the terms that involve the variable \(x\) because they are considered like terms. This gives \((1x + 4x - 5x + 1x + 4x + 1x) + (4 - 2)\).

• The coefficients (numbers in front of the variables) are added or subtracted according to their sign.
• For instance, \(1 + 4 - 5 + 1 + 4 + 1 = 6\), so we have \(6x\).
• Similarly, constant terms (numbers without variables) are added or subtracted to combine them: \(4 - 2 = 2\).

This leaves us with a simpler expression: \(6x + 2\). Combining like terms is fundamental as it sets up a simpler equation for further solving.

Simplifying Equations

Simplifying equations helps to make them easier to solve. You simplify an equation by performing operations such as combining like terms or clearing out unnecessary components. For example, after combining like terms in the provided equation, we obtained \(6x + 2 = -3x - 8\).

• Now, to simplify this equation further, you would add \(3x\) to both sides to isolate the variable \(x\) on one side: \(9x + 2 = -8\).
• Next, you remove the constant term by subtracting 2 from both sides, resulting in \(9x = -10\).

Each step in simplifying brings clarity, reducing the elements you need to work with, which leads to a straightforward solution.

Multiplication and Division in Equations

Handling multiplication and division properly is crucial when solving equations. Each operation must be performed equally across the equation to maintain balance.

• Initially, interpret expressions like \(2x\) as multiplication: \(2 \, \cdot \, x\).
• When equations involve multiplication, such as \(2(x \cdot 2)\), this calculates to \(4x\).
• To solve equations after adding or subtracting terms, you may need to divide. Consider \(9x = -10\); dividing both sides by 9 gives \(x = -\frac{10}{9}\).

Remember, dividing by a number means splitting the total into that many equal parts. Doing equal operations on both sides preserves equality, ensuring the solution is valid.