Problem 78
Question
Solve the equation and check your solution. (Some of the equations have no solution.) $$\frac{3}{4}+5 s=-2+5 s$$
Step-by-Step Solution
Verified Answer
The equation, \(\frac{3}{4}+5 s = -2+5 s\), doesn't have any solution.
1Step 1: Organize the equation
We start by organizing the equation. We have the term \(5s\) on both sides of the equation, which simplifies to: \[\frac{3}{4} = -2,\] once we subtract \(5s\) from both sides.
2Step 2: Solve for the equation
The step next is to further simplify the equation. In its current state, it is clear that \(\frac{3}{4} \neq -2\), hence the equation does not hold true.
3Step 3: Check if the equation has any solution
Since \(\frac{3}{4} \neq -2\), the equation doesn't have any root. Thus, we conclude that there is no solution for this equation.
Key Concepts
Equation SolvingAlgebraic ManipulationNo Solution Equations
Equation Solving
Solving an equation involves finding the value of the unknown that makes the equation true. An equation acts like a balance scale; what you do to one side, you should do to the other to maintain balance. In the given problem, the equation is \( \frac{3}{4} + 5s = -2 + 5s \).
To begin solving, we identify the terms involving the variable \( s \), which appear on both sides of the equation. A useful first step is to move all like terms to opposite sides.
By subtracting \( 5s \) from both sides, we simplify the equation to \( \frac{3}{4} = -2 \). This manipulation makes it easier to focus on just the numerical aspect, setting aside the variable in this instance. Equation solving sometimes reveals unexpected results, such as a situation where no solution exists.
To begin solving, we identify the terms involving the variable \( s \), which appear on both sides of the equation. A useful first step is to move all like terms to opposite sides.
By subtracting \( 5s \) from both sides, we simplify the equation to \( \frac{3}{4} = -2 \). This manipulation makes it easier to focus on just the numerical aspect, setting aside the variable in this instance. Equation solving sometimes reveals unexpected results, such as a situation where no solution exists.
Algebraic Manipulation
Algebraic manipulation is used to rearrange and simplify equations. It's like organizing pieces of a puzzle to see the complete picture. In this exercise, algebraic manipulation involves simplifying \( \frac{3}{4} + 5s = -2 + 5s \).
This process uses basic principles, such as adding, subtracting, or dividing both sides of the equation by the same amount. Our goal is to isolate the variable or highlight an inherent contradiction.
After moving \( 5s \) from both sides, the equation \( \frac{3}{4} = -2 \) is derived. Through this algebraic step, you can see if the numbers logically equate, allowing for straightforward solutions or, as in this case, indicating an inconsistency. Such manipulations are crucial in solving much more complex algebraic equations.
This process uses basic principles, such as adding, subtracting, or dividing both sides of the equation by the same amount. Our goal is to isolate the variable or highlight an inherent contradiction.
After moving \( 5s \) from both sides, the equation \( \frac{3}{4} = -2 \) is derived. Through this algebraic step, you can see if the numbers logically equate, allowing for straightforward solutions or, as in this case, indicating an inconsistency. Such manipulations are crucial in solving much more complex algebraic equations.
No Solution Equations
Sometimes, equations don’t yield any solution. This happens when, after simplifying, we end up with a nonsensical statement, such as \( \frac{3}{4} = -2 \).
Recognizing a no solution equation is just as important as solving one with a solution. These scenarios occur typically when variables cancel out, showing that no real number satisfies the original equation.
In this example, the final statement indicates that the relationship between both sides never holds true. It's vital to check work through simplification processes to ascertain whether an equation has no solution, ensuring every step uncovers truths about the calculations involved. It is a reminder that not all mathematical problems have straightforward outcomes.
Recognizing a no solution equation is just as important as solving one with a solution. These scenarios occur typically when variables cancel out, showing that no real number satisfies the original equation.
In this example, the final statement indicates that the relationship between both sides never holds true. It's vital to check work through simplification processes to ascertain whether an equation has no solution, ensuring every step uncovers truths about the calculations involved. It is a reminder that not all mathematical problems have straightforward outcomes.
Other exercises in this chapter
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