Problem 11 - Elementary Algebra

Question

Solve. $$ -4 x+2=7 x-20 $$

Step-by-Step Solution

Answer

The solution is \(x = 2\).

Step 1: Isolate the Variable Terms

Start by rearranging the equation so that all terms containing the variable \(x\) are on one side of the equation. Move \(7x\) by subtracting \(7x\) from both sides of the equation:\[-4x + 2 - 7x = 7x - 7x - 20\]This simplifies to:\[-11x + 2 = -20\]

Step 2: Move the Constant Term to the Opposite Side

Next, isolate the term with \(x\) by moving the constant on the left side of the equation to the right. Subtract \(2\) from both sides:\[-11x + 2 - 2 = -20 - 2\]This simplifies to:\[-11x = -22\]

Step 3: Solve for the Variable

Now solve for \(x\) by dividing both sides of the equation by \(-11\):\[x = \frac{-22}{-11}\]This simplifies to:\[x = 2\]

Step 4: Verify the Solution

Substitute \(x = 2\) back into the original equation to verify the solution:\[-4(2) + 2 = 7(2) - 20\]Calculate each side:\[-8 + 2 = 14 - 20\], which simplifies to \[-6 = -6\].Since both sides are equal, the solution is verified.

Key Concepts

Variable Isolation

When solving linear equations, the goal is often to isolate the variable, which in our example is 'x'. This means getting all terms that involve 'x' on one side of the equation. Imagine splitting the equation into two sides, just like balancing scales. To keep the balance, you need to perform the same operation on both sides. In our example, we start by moving 7x from the right to the left by subtracting it from both sides:

• Original: \[-4x + 2 = 7x - 20\]
• After moving: \[-4x + 2 - 7x = -20\]
• Simplified: \[-11x + 2 = -20\]

Once 'x' is isolated among other terms on one side, it becomes easier to handle. Clustering like terms together is crucial because it sets the stage for further simplification.

Constants

Constants are the numbers in an equation that do not change, unlike variables which can vary. In our equation, '2' and '-20' are constants. After isolating terms involving 'x', the next logical step is to handle the constants. By moving the constant term on the side of 'x' across the equation, we get:

• Start: \[-11x + 2 = -20\]
• Move 2: \[-11x + 2 - 2 = -20 - 2\]
• Result: \[-11x = -22\]

This relocation helps us in simplifying the equation as it fully uncouples 'x' from numbers without 'x', leaving us a cleaner equation to solve for 'x'.

Equation Verification

Once a value for 'x' is found, it's necessary to check if it truly is a solution of the initial equation. This is done through equation verification. By substituting the solved value of 'x' back into the original equation, we can check if both sides remain equal:

• Substituting: \[-4(2) + 2 = 7(2) - 20\]
• Calculating, simplify \[-8 + 2 = 14 - 20\]
• Result: \[-6 = -6\]

Since both equations balance with the value 'x = 2', this confirms our solution is correct. Verification helps ensure there were no errors during solving.

Simplification of Equations

Simplification is about making equations less complicated. This involves combining like terms and using elementary operations to condense the equation to its simplest form. In our solved equation, this process looked like:

• \[-4x + 2 - 7x = 7x - 7x - 20\]
• Combined like terms: \[-11x + 2 = -20\]
• After moving constants: \[-11x = -22\]
• Solved for x: \[x = \frac{-22}{-11} = 2\]

Simplifying was basically peeling away layers of complexity to ultimately find the value of 'x'. It involves clear steps that are easily followed to avoid mistakes.