Problem 58 - Elementary Algebra

Question

Solve. $$ 35 x-12=110 $$

Step-by-Step Solution

Answer

The solution is \( x \approx 3.49 \).

Step 1: Isolate the Variable Term

To solve the equation \( 35x - 12 = 110 \), first isolate the term containing the variable (\(35x\)). We do this by adding 12 to both sides of the equation. \[ 35x - 12 + 12 = 110 + 12 \]This simplifies to:\[ 35x = 122 \]

Step 2: Solve for the Variable

Now that we have \(35x = 122\), solve for \(x\) by dividing both sides of the equation by 35. \[ x = \frac{122}{35} \]This gives us:\[ x \approx 3.49 \]

Key Concepts

Solving Equations

Solving an equation means finding the value of the unknown variable that makes the equation true. It’s like figuring out a mystery number that balances everything perfectly. Equations involve expressions that have equal signs, indicating that what’s on one side has the same value as what’s on the other. Using operations such as addition, subtraction, multiplication, or division, we try to simplify the equation.

For example, in the equation \(35x - 12 = 110\), our goal is to find out what \(x\) is. Solving such equations is an essential skill because it's the foundation for more complex problem-solving in math and science. The primary aim is to perform actions that simplify either side, ensuring the equation stays balanced, eventually leading you to the value of the variable. Remember, the final answer should validate the original equation when substituted back.

Isolate Variable

Isolating the variable is a key step when solving an equation because it simplifies the task into a more manageable one. The idea is to get the variable on one side of the equation by itself. This means eliminating any number or coefficient that is on the same side as the variable.

To isolate the variable in the equation \(35x - 12 = 110\), we first remove the \(-12\) by adding 12 to both sides of the equation. This is because of the rules of balancing that allow us to manipulate equations by performing the same operation on both sides:

• Addition or subtraction (e.g. adding 12 both sides).
• Multiplication or division (necessary in our next step).

After these steps, \(35x =122\), and our equation becomes simpler with \(x\) closer to being isolated. This is the power of isolating variables; it eventually leads us directly towards solving the equation.

Linear Equations

A linear equation is an equation that makes a straight line when graphed. It is an equation of the first degree, which means it has no exponents higher than one and no variables in the denominator. Linear equations typically look like \(ax + b = c\), where \(a\), \(b\), and \(c\) are real numbers.

In this exercise, \(35x - 12 = 110\) is an example of a linear equation. The characteristics of this equation include:

• Having one variable, \(x\).
• A constant term \(-12\).
• A coefficient \(35\) for \(x\).

The steps to solve it are straightforward, involving isolating the variable and simplifying further. Linear equations are vital because they simplify understanding relationships and making predictions in various real-world scenarios, like calculating costs, predicting growth, and understanding rates. By mastering linear equations, you build a solid foundation for more advanced math topics.