Problem 10 - Prealgebra

Question

Solve each equation using the methods shown in this section. $$15 x+1=-4 x+20$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 1\).

Step 1: Combine like terms

Begin by adding \(4x\) to both sides of the equation to get all \(x\) terms on one side:\[15x + 1 + 4x = 20\]This simplifies to:\[19x + 1 = 20\]

Step 2: Isolate the variable term

Subtract 1 from both sides to isolate the \(x\) term:\[19x = 20 - 1\]This simplifies to:\[19x = 19\]

Step 3: Solve for \(x\)

Divide both sides by 19 to solve for \(x\):\[\frac{19x}{19} = \frac{19}{19}\]This simplifies to:\[x = 1\]

Key Concepts

Combining Like Terms

When solving linear equations, one of the first steps to simplify an equation is by combining like terms. This means grouping together terms that have the same variable part. In our example, the equation starts as: \[15x + 1 = -4x + 20\]
Begin by moving the terms involving \(x\) to one side of the equation. You do this by adding \(4x\) to both sides. This helps in combining the like terms: \[15x + 4x + 1 = 20\]
On combining, you find: \[19x + 1 = 20\] This step aims to have all \(x\) terms on the same side, simplifying the equation and preparing it for the next steps. Always remember, only like terms - those with the exact same variable and exponent - can be combined. This approach will make your equations much easier to handle.

Isolation of Variables

Once like terms are combined, the next objective is to isolate the variable term. This essentially means getting the variable \(x\) alone on one side of the equation. After combining the \(x\) terms as shown earlier, you get: \[19x + 1 = 20\]
To isolate \(19x\), you will subtract 1 from both sides. This operation removes the constant from the left side:
- Subtract 1 from both sides to get: \[19x = 20 - 1\] This results in: \[19x = 19\] By achieving this, you focus on the single \(x\) term by eliminating other numbers that are grouped with it. Isolation involves using inverse operations like subtraction here, where the idea is always to perform the reverse of the operation present.

Simplifying Equations

The final step in solving the linear equation is all about simplification. With the equation \(19x = 19\), we aim to find the value of \(x\) by simplifying further. This is often where we actually 'solve' the equation.
To complete this step, divide both sides of the equation by 19, which is the coefficient of \(x\): \[\frac{19x}{19} = \frac{19}{19}\] This simplification gives you: \[x = 1\] Here, division is used because it is the opposite of multiplication. Hence, it cancels out the coefficient (19) in front of \(x\), leaving \(x\) by itself. In any equation solving process, the goal of simplification is always to reduce the equation to a simple expression of the variable on one side.