Problem 60 - Introductory Algebra for College Students

Question

Solve each equation. Use words or set notation to identify equations that have no solution, or equations that are true for all real numbers. $$2(x-5)=2 x+10$$

Step-by-Step Solution

Answer

The equation has no solution, because rearranging and validating the equation yields an inequality (-10 ≠ 10).

Step 1: Distribute Coefficient

Multiply the term outside the brackets with each term within the brackets to simplify. This results in \(2x - 10 = 2x + 10\).

Step 2: Rearrange the Equation

Subtract \(2x\) from both sides of the equation. This gives: \(-10 = 10\).

Step 3: Validate the Equation

\(-10 = 10\) is clearly not true, as -10 equals -10 and 10 equals 10, but -10 does not equal 10. Therefore, the equation is invalid and has no solution.

Key Concepts

No Solution

When we talk about an equation having no solution, it means that there is no possible value that can be substituted for the variable to make the equation true. In simpler terms, however hard you try, there’s no number out there that will satisfy the equation. Imagine you have a balance scale, and no matter what weight you place on one side, both sides will not level out.

In the example provided, \[2(x-5) = 2x + 10\]distributing the terms gives:\[2x - 10 = 2x + 10\].
When we subtract \(2x\) from both sides, we end up with:\[-10 = 10\]. This equation doesn’t make sense because -10 is not equal to 10. It's like trying to say that a cat is the same as a dog; it's simply not true. Therefore, in cases like these, the equation has no solution.

Recognizing a no solution equation helps in making quick judgments, especially when solving many equations.

Distributive Property

The distributive property is a fundamental algebraic principle that allows us to multiply a single term across terms within parentheses. This operation is key in simplifying equations, and it makes complex expressions easier to handle. Think of it like distributing candies equally among children so that everyone gets a share.

Given a simple example:\[a(b + c) = ab + ac\], the term outside the parentheses \(a\) gets multiplied by both \(b\) and \(c\). It's important because this step is usually among the first in simplifying expressions.

• Helps simplify and solve equations.
• Vital for expanding expressions.
• Ensures that multiplication is applied correctly to each term within the parentheses.

With our exercise starting as \[2(x-5) = 2x + 10\], the term outside the parentheses, \(2\), must be multiplied by both \(x\) and \(-5\), resulting in \(2x - 10\).

Remembering this step guides you through many algebra problems efficiently.

Equivalent Equations

Equivalent equations are different equations that have the same set of solutions. It's like having different keys that all open the same door. When you manipulate equations by doing valid operations like adding, subtracting, multiplying, or dividing both sides by the same non-zero number, the equations remain equivalent.

For example, \[x + 2 = 5\] and \[x = 3\] are equivalent because both have the solution \(x = 3\).

• Always maintain the balance of equality by applying the same operation on both sides.
• Used to simplify equations and make solving easier.
• Helps in understanding different forms of the same equation.

In our original problem, after rearranging, the equation wasn’t equivalent because \(-10 = 10\) doesn’t hold any possible solutions.

Grasping equivalent equations is significant because it shows the path from a complicated equation to its simplest form while ensuring that no solutions are lost along the way.