Problem 34 - Algebra Form and Function

Question

Does the equation have no solution, one solution, or an infinite number of solutions? $$ 4 x+3=4(x-1)+5 $$

Step-by-Step Solution

Answer

Question: Determine the type of solution in the given equation: $$4x + 3 = 4(x - 1) + 5$$ Answer: No solution

Step 1: Distribute the 4 on the right side of the equation

Multiply 4 by both terms inside the parentheses: $$4x + 3 = 4x - 4 + 5$$

Step 2: Simplify the equation

Combine the constants on the right side of the equation: $$4x + 3 = 4x + 1$$

Step 3: Subtract 4x from both sides of the equation

Eliminate the x term from one side of the equation by subtracting 4x: $$3 = 1$$

Step 4: Analyze the result

Since we are left with an equation that states "3 = 1," which is not true, it means there is no solution for the given equation.

Key Concepts

Equation solving

Equation solving is the process of finding the value of the variable that makes an equation true. It involves several techniques and strategies to isolate the variable and gain insight into the equation's behavior. To start solving an equation, you need to identify what type of operations have been used, such as addition, subtraction, multiplication, or division. The aim is to perform inverse operations to simplify and isolate the variable step by step.

• Always start by simplifying both sides of the equation if possible.
• Next, use inverse operations to move terms from one side of the equation to the other.
• Your goal is to have the variable alone on one side of the equation for easy solving.

In the example from the exercise, the solution process involved simplifying the expressions and cancelling out terms systematically. This approach is fundamental in algebra, where each step builds toward an answer, or sometimes reveals there is no solution at all.

Distributive property

The distributive property is a valuable algebraic tool used to simplify expressions and solve equations effectively. It states that multiplying a single term by a sum inside parentheses is equivalent to multiplying the term by each addend and then adding the results. Mathematically, it is expressed as:\[ a(b + c) = ab + ac \] In simpler terms, if you have a number outside the parentheses, you distribute, or "pass out," the number through multiplication across each term inside the parentheses. In our exercise, we saw this property in action with the equation:\[ 4(x-1) \] Here, the number 4 is multiplied by both \( x \) and \(-1\), resulting in \(4x -4\). This step simplifies and prepares the equation for further solving. Mastery of the distributive property helps in tidying up expressions and is a staple part of algebraic manipulations.

No solution in equations

A no solution equation arises when, after simplification, we reach a false statement. This means that no real number will satisfy the equation and make it true. In our solved exercise:\[4x + 3 = 4x + 1\] After subtracting \(4x\) from both sides, we ended up with:\[3 = 1\] Since 3 will never equal 1, it's clear the equation has no solution. This kind of result often happens when the variables cancel out completely, leaving behind a contradiction.

• Equations are balanced scales; if balance can't be restored due to a contradiction, there's no solution.
• Recognizing no solution saves time since it alerts us early on that solving further is futile.

Equations without solutions teach us about the system's limitations they represent and how some assumptions might not have a valid outcome.