Problem 23 - Algebra A Combined Function

Question

Solve each equation. Don't forget to first simplify each side of the equation, if possible. Check each solution. See Examples 5 through 7 . $$ 5 x-6=6 x-5 $$

Step-by-Step Solution

Answer

The solution is \(x = -1\).

Step 1: Simplify Both Sides of the Equation

The equation is already simplified as there are no like terms to combine on either side: \(5x - 6 = 6x - 5\).

Step 2: Move Variable Terms to One Side

Subtract \(5x\) from both sides of the equation to get the variable terms on one side:\(5x - 6 - 5x = 6x - 5 - 5x\), which simplifies to \(-6 = x - 5\).

Step 3: Isolate the Variable

Add 5 to both sides to isolate \(x\):\(-6 + 5 = x - 5 + 5\), which simplifies to \(-1 = x\).

Step 4: Check the Solution

Substitute \(x = -1\) back into the original equation to verify:\(5(-1) - 6 = 6(-1) - 5\).Simplifying both sides gives:\(-5 - 6 = -6 - 5\) or \(-11 = -11\).The left side equals the right side, confirming that \(x = -1\) is the correct solution.

Key Concepts

Simplifying Equations

When you solve linear equations, the first crucial step is simplifying. This means you need to look at each side of the equation to see if any terms can be combined or if there is any need to simplify before proceeding further. For example, if you have a term like \(2x + 3x\), you can combine these to get \(5x\). Likewise, if there are constants that can be added or subtracted, you should do so.In the exercise provided, the expression \(5x-6=6x-5\) appears initially as one that doesn't require further simplification. This is because there are no like terms on each side of the equation to combine.Remember:

• Always check for like terms on each side of the equation.
• Combine like terms to simplify your expressions, but only on each individual side.
• Simplification sets up the equation to make further steps like moving variables or isolating them easier.

By simplifying first, you ensure clarity in solving the equation effectively.

Isolating Variables

Once you've simplified, the next focus is isolating the variable. Isolating the variable means rearranging the equation such that the variable (often \(x\) in linear equations) is on one side and everything else on the other.In the example \(5x - 6 = 6x - 5\), you'll need to move all terms involving \(x\) to one side. This is done by subtracting \(5x\) from both sides, which gives us \(-6 = x - 5\).Tips for isolating variables:

• Perform the same operation to both sides of the equation to keep it balanced.
• Look to simplify further after moving variables to reduce the equation to an isolated state \(x = ...\).
• Don't be afraid to add or subtract numbers to get the variable alone on one side.

Achieving this step well sets you up perfectly to solve for the exact value of the variable.

Checking Solutions

It's always smart to check your work by verifying your solution. This involves substituting the value you found back into the original equation to ensure both sides equal.For the equation \(5x-6=6x-5\) and solution \(x=-1\), substitute \(-1\) back in:\[5(-1) - 6 = 6(-1) - 5\]Simplifying both sides results in:\[-5 - 6 = -6 - 5\] which becomes \(-11 = -11\).Helpful checking hints:

• Always substitute back into the original, not the simplified version, to catch any errors in your steps.
• Both sides must be equal for the solution to be correct.
• Consider checking solutions even when sure; it solidifies understanding and confirms accuracy.

This final verification reinforces the correctness of your solution and boosts confidence in your method.