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Chapter 1: Problem 4

solve and check each linear equation. $$ 5 x-(2 x-10)-35 $$

Short Answer

Expert verified
The solution to the equation is \(x = \frac{25}{3}\)

Step by step solution

01

Distribute the negative

First, distribute the negative sign across the term \(-(2 x-10)\). This changes the problem to: \(5x - 2x + 10 - 35\)
02

Combine like-terms

Now, combine the like terms (terms that have variable x and the constants). This changes the problem to: \(3x - 25\)
03

Isolate x

To isolate x, add 25 to both sides of the equation. This changes the equation to: \(x = \frac{25}{3}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a key concept in algebra that helps to simplify expressions. It essentially allows you to remove parentheses by distributing a factor across terms within the parentheses. This can be thought of as sharing whatever is outside of the parentheses with everything inside. For example, let's consider the expression \(5x - (2x - 10)\). Here, we need to distribute the negative sign over the terms inside the parentheses.
  • Distribute the negative sign: \(-(2x - 10)\) becomes \(-2x + 10\).
  • Rewrite the original expression: \(5x - (2x - 10)\) becomes \(5x - 2x + 10\).
By applying the distributive property, you can effectively eliminate parentheses and simplify the equation, making it easier to solve in subsequent steps.
Combining Like Terms
After using the distributive property, you often end up with an expression containing like terms that you can combine. Like terms are terms that have the same variable raised to the same power, or are simply constants.
  • In our expression \(5x - 2x + 10 - 35\), the like terms include \(5x\) and \(-2x\), both involving the variable \(x\).
  • Combine these like terms: \(5x - 2x = 3x\).
  • There are also constant terms \(+10\) and \(-35\), which combine to form \(-25\).
So, combining like terms changes the expression to \(3x - 25\). This simplification process is crucial in solving equations as it reduces complexity and makes finding the solution much more straightforward.
Solving Equations
Once the expression is simplified by distributing and combining like terms, the next step is solving the equation. Solving an equation involves finding the value of the variable that makes the equation true. Here鈥檚 how to solve for \(x\) in our simplified equation \(3x - 25\).
  • First, isolate the term containing the variable on one side of the equation. Start by adding 25 to both sides to cancel out the \(-25\):
    \(3x - 25 + 25 = 0 + 25\), resulting in \(3x = 25\).
  • Next, divide both sides by the coefficient of \(x\) (which is 3) to solve for \(x\):
    \(x = \frac{25}{3}\).
This process of isolating the variable is essential in solving equations because it allows you to find the exact value that the variable represents. It is a fundamental skill in algebra that is used in more complex problems and different areas of mathematics.

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Most popular questions from this chapter

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