Question

Solve: \(60 x-900=-15 x+2850 .\)

Short answer

x=50

Step-by-step solution

- Combine like terms

First, we need to get all terms involving the variable on one side of the equation and the constant terms on the other side. We start by adding \(15x\) to both sides: \(60x + 15x - 900 = 2850\).

- Simplify the equation

Next, we combine the like terms on the left side: \(60x + 15x = 75x\). This gives us the simplified equation: \(75x - 900 = 2850\).

- Isolate the variable term

Add \(900\) to both sides of the equation to isolate the term with the variable: \(75x - 900 + 900 = 2850 + 900\). This simplifies to \(75x = 3750\).

- Solve for the variable

Finally, divide both sides by \(75\) to solve for \(x\): \(\frac{75x}{75} = \frac{3750}{75}\). Thus, \(x = 50\).

Key concepts

Combining Like Terms

Combining like terms is an essential step in simplifying linear equations.
Like terms are terms that contain the same variable raised to the same power.
For instance, in the expression \(60x - 900 = -15x + 2850\), the terms \(60x\) and \(-15x\) are like terms because they both contain the variable \(x\).
Similarly, \(-900\) and \(2850\) are constants, making them like terms as well.
To combine like terms, we move terms with variables to one side and constants to the other.
This often involves adding or subtracting terms on both sides of the equation to maintain balance.
In our example, adding \(15x\) to both sides gives us: \(60x + 15x - 900 = 2850\).
This step simplifies the equation and sets the stage for the next steps.

Isolating Variables

The core idea behind isolating variables is to have the variable term stand alone on one side of the equation.
This is done by using inverse operations to 'undo' other operations surrounding the variable.
In our example, after combining like terms, we had: \(75x - 900 = 2850\).
To isolate the variable term \(75x\), we add \(900\) to both sides: \(75x - 900 + 900 = 2850 + 900\).
This simplifies to \(75x = 3750\).
Isolating variables is a critical technique in solving equations because it allows us to see what the variable equals directly.

Linear Equations

A linear equation is an equation where each term is either a constant or the product of a constant and a single variable.
Linear equations look like this: \(ax + b = c\).
In our case, \(60x - 900 = -15x + 2850\), we need to rearrange it to fit the standard linear equation form.
By applying the steps explained, we isolate the variable, making it easier to solve for it.
Linear equations are foundational in algebra because they represent a straight line when graphed.
Solving linear equations forms the basis for more complex algebraic concepts.

Algebraic Manipulation

Algebraic manipulation involves using algebraic rules and properties to transform and solve equations.
It includes operations like addition, subtraction, multiplication, division, as well as factoring and expanding expressions.
For our exercise, we used several manipulations:

• Combining like terms: Added \(15x\) to each side
• Simplifying terms: Combined \(60x + 15x\) into \(75x\)
• Isolating the variable: Add \(900\) to remove the constant term
• Solving: Divide both sides by \(75\) to find \(x = 50\)

These steps show how algebraic manipulation is crucial for simplifying and solving equations accurately.
Mastery of these techniques makes solving algebra problems straightforward and intuitive.