Problem 69 - Algebra 1

Question

Solve the equation. $$ 4 x-11=-31 $$

Step-by-Step Solution

Answer

The solution to the equation \4x - 11 = -31\ is \(x = -5\)

Step 1: Add 11 to Both Sides

Taking the equation \(4x - 11 = -31\), add 11 to both sides to isolate the term with x on one side. This simplifies to \(4x = -31 + 11\)

Step 2: Simplify the Right Side

Simplify the right side of the equation by performing the addition: \(-31 + 11 = -20\). This reduces the equation to \(4x = -20\)

Step 3: Divide Both Sides by 4

To further isolate x, divide both sides of the equation by 4: \(x = -20 / 4\)

Step 4: Simplify to Get the Value of x

Perform division to get the value of x: \(x = -5\)

Key Concepts

Addition Property of Equality

The addition property of equality is a fundamental principle in solving linear equations. Simply put, it states that if two quantities are equal, then adding the same number to both sides of the equation does not change their equality. This is crucial in maintaining balance in an equation.
For example, in the equation:

• 4x - 11 = -31

Our goal is to isolate the variable term. By using the addition property of equality, we add 11 to both sides to eliminate the -11. This step simplifies our equation and assists in isolating the variable more effectively.
Thus, the equation becomes:

• 4x - 11 + 11 = -31 + 11
• 4x = -20

By adding 11 to each side, we successfully remove the constant term next to 4x.

Isolation of Variables

Isolation of variables is the method used to manipulate an equation in a way that leaves the variable by itself on one side. This is essential for solving the equation for the unknown variable.
After applying the addition property of equality in the previous step, we simplify the task by focusing on the variable term:

• 4x = -20

Here, the variable x is not completely isolated as it is multiplied by 4. To isolate x fully, further steps are required such as division.
Isolating a variable involves reversing operations that are keeping the variable from standing alone. Always follow the rule of performing the same operations to both sides to maintain a balanced equation.

Division

Division is the process used to completely isolate the variable in an algebraic equation. In our example, once we arrive at the equation where 4x = -20, division helps achieve the final step of isolating x.
Since x is multiplied by 4, dividing both sides by 4 will remove this multiplication:

• \( \frac{4x}{4} = \frac{-20}{4} \)
• \( x = -5 \)

Division is especially useful when you have a coefficient multiplying the variable. By simply dividing, we effectively bring the coefficient to "1," thus isolating x. This operation provides us with the exact value of the variable, which is -5 in this solution.
In summary, division is a powerful tool to use alongside techniques such as the addition property of equality to solve equations systematically.