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Chapter 8: Problem 4

Solve each equation. Be sure to note whether the equation is quadratic or linear. $$x+4=5 x$$

Short Answer

Expert verified
Linear equation. The solution is x = 1.

Step by step solution

01

- Identify the Type of Equation

Notice that the given equation is linear because the variable x is not raised to any power greater than 1.
02

- Move All Terms Involving x to One Side

Subtract x from both sides of the equation to collect all terms involving x on one side: x + 4 - x = 5x - x This simplifies to: 4 = 4x
03

- Isolate x

Divide both sides of the equation by 4 to solve for x: \( x = \frac{4}{4} \) This simplifies to: \( x = 1 \)

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Key Concepts

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

linear equations
A linear equation is an algebraic equation in which the highest power of the variable is one. This is characterized by expressions like Ax + B = 0, where x is the variable. Linear equations are straightforward as they form a straight line when graphed.
In our example: $$x + 4 = 5x$$, the equation is linear because the variable x is raised to the power of one (no exponents other than 1). Spotting a linear equation is essential for knowing what steps to take to solve it.
isolate variable
Isolating the variable means rearranging the equation to get the variable alone on one side. This usually involves:
  • Adding or subtracting terms to move them to the opposite side.
  • Dividing or multiplying to get the coefficient of the variable to be one.
In our example, the original equation is $$x + 4 = 5x$$. To isolate x:
1. Move all x terms to one side: $$x + 4 - x = 5x - x$$ simplifies to $$4 = 4x$$.
2. Divide both sides by 4 to isolate x: $$x = \frac{4}{4}$$ which simplifies to $$x = 1.$$
step-by-step solution
Solving equations methodically ensures accuracy. With our linear equation $$x + 4 = 5x$$, follow these steps:
1. **Identify the equation type:** It's linear because x is not raised to any power higher than 1.
2. **Move terms involving x to one side:** Subtract x from both sides: $$x + 4 - x = 5x - x$$, simplifying to $$4 = 4x$$.
3. **Isolate x:** Divide both sides by 4: $$x = \frac{4}{4}$$, which simplifies to $$x = 1$$.
Following these steps ensures you systematically address all parts of the equation.

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Most popular questions from this chapter

Express the given number in scientific notation. $$0.00431$$

Look at the following and discuss what is happening at each step. $$\begin{array}{l}x^{2}-x-20 \\\\\quad=x^{2}-5 x+4 x-20 \\\\\quad=x(x-5)+4(x-5) \\\\\quad=(x+4)(x-5)\end{array}$$

Factor each of the following expressions as completely as possible. If an expression is not factorable, say so. $$t^{2}+1$$

In a basketball league of \(n\) teams in which each team plays every other team twice, the total number of games played is \(n^{2}-n .\) How many teams are there in a league that plays a total of 90 games?

Divide each of the following. Use the long division process where necessary. $$\frac{x^{3}-5 x+4}{x-1}$$
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