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Chapter 5: Problem 70

Solve the following equations. See Section 2.1 $$ 5 x-15=0 $$

Short Answer

Expert verified
The solution is \(x = 3\).

Step by step solution

01

Understanding The Equation

Given the equation \(5x - 15 = 0\), we need to find the value of \(x\). This is a simple linear equation where we need to solve for \(x\).
02

Add 15 to Both Sides

To isolate the term with \(x\), start by eliminating the constant on the left side. Add 15 to both sides of the equation: \[ 5x - 15 + 15 = 0 + 15 \]which simplifies to:\[ 5x = 15 \]
03

Divide Both Sides By 5

Now, divide both sides of the equation by 5 to solve for \(x\): \[ \frac{5x}{5} = \frac{15}{5} \] which simplifies to: \[ x = 3 \]

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Key Concepts

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

Solving Equations
Solving equations is a fundamental process in algebra that involves finding the value of a variable that makes the equation true. In our example, the equation is \(5x - 15 = 0\). Here, we need to determine the value of \(x\) that satisfies this equation. The primary aim while solving equations is to perform operations that maintain balance. This can mean adding, subtracting, multiplying, or dividing both sides by the same number. When you solve an equation, you're essentially finding the equilibrium point where both sides are equal. This approach is a basis for solving more complex equations as well.
Step-by-Step Solutions
A step-by-step solution provides a clear path to solving an equation, breaking it down into manageable parts. In our example, the initial step is to see \(5x - 15 = 0\) and recognize that we want \(x\) to stand alone on one side of the equation.
  • Step 1: Identify the structure of the equation. This includes knowing what's asked and what operations are presented鈥攈ere, subtraction and multiplication.
  • Step 2: Remove any constants from the side with the variable by performing the opposite operation. For instance, to remove \(-15\), we add 15 to both sides.
  • Step 3: Simplify the equation after each operation. This allows us to progressively get closer to the solution.
  • Step 4: Isolate the variable. This often involves dividing or multiplying steps, as done here by dividing both sides by 5.
Using these structured steps simplifies complex problems and makes them easier to solve.
Isolating Variables
Isolating variables is a critical technique to solve linear equations efficiently. When you isolate a variable, you're simplifying the equation so that the variable is by itself on one side. In the equation \(5x - 15 = 0\), our goal is to isolate \(x\). To achieve this:
  • First, eliminate any constants from the side with the variable. Here, we added 15 to both sides to cancel out \(-15\).
  • Next, we deal with any coefficients attached to the variable by performing the inverse operation. Since \(x\) is multiplied by 5, we divide both sides by 5 to isolate \(x\).
Comprehending how to isolate variables is necessary, as it builds the foundation for solving more advanced algebraic equations. It is important to remember that every operation should maintain the balance of the equation.

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Most popular questions from this chapter

Factor each polynomial completely. See Examples 1 through 12. $$ 3 x^{2}-6 x+3 $$

Evaluate. See Sections 1.3 and 5.1 $$ \left(2^{2}\right)^{3} $$

If \(P(x)=3 x+3, Q(x)=4 x^{2}-6 x+3,\) and \(R(x)=5 x^{2}-7\) find the following. \(-5[P(x)]-Q(x)\)

Factor each trinomial. See Examples 5 through 10. $$ 28 y^{2}+22 y+4 $$

Factor each polynomial completely. See Examples 1 through 12. $$ 36 x y^{2}-48 x y z^{2}+16 x z^{4} $$
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