Problem 105 - Algebra and Trigonometry

Question

determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 6+\frac{1}{x}=\frac{7}{x} $$

Step-by-Step Solution

Answer

The original statement is false. The correct equation would be \(6 + \frac{1}{x} = -6x\).

Step 1: Isolate the fraction value on the left side

Subtract 6 from both sides of the equation. Which gives you: \(\frac{1}{x}= \frac{7}{x}-6\)

Step 2: Find the least common denominator(LCD) and combine the fractions

The denominators are 'x' and 'x' which are the same so they can be combined directly. This results in: \(\frac{1}{x}=-6\). The equation simplifies further to: \(x*\frac{1}{x}=-6*x\).

Step 3: Simplify the equation to show the falsity of the original statement

The equation simplifies to: \(1=-6*x\) . If you solve for \(x\), you still will not get an answer such that the original equation is true. So, the original equation is indeed false.

Step 4: Change original false equation to make a true equation

Since we know \(6 + \frac{1}{x} = -6x\), we can replace \(\frac{7}{x}\) with \( -6x\) to produce a correct equation. Thus, \(6 + \frac{1}{x} = -6x\) is the correct equation.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create straight lines when graphed. They are fundamental in algebra and are typically set in the form of \(ax + b = c\). Understanding linear equations helps in solving real-world problems, which makes them essential in mathematics.
What distinguishes a linear equation from other types? It is the absence of exponents on variables, meaning every term should be either a constant or a variable raised to the power of one. This is why they stay linear rather than becoming curved like quadratic equations or exponential functions.

• Linear equations usually involve two operations: addition/subtraction and multiplication/division.
• They are used to find an unknown variable that produces a solution on both sides of the equation.
• The variables are typically denoted as \(x\) or \(y\) and are combined with constants to form equations.

Learning how to manipulate these equations is essential since they form the foundation of algebraic concepts. You can isolate terms to find solutions and even use substitution or elimination methods when dealing with systems of equations.

Solving Equations

Solving equations means finding the value of a variable that makes the equation true. It's like reverse engineering the equation to locate the unknown value. This process generally involves a series of steps to isolate the variable on one side.
Here are some steps you might follow when solving an equation:

• Identify and simplify both sides where possible.
• Use inverse operations to eliminate numbers attached to the variable.
• Check your solution by substituting back into the original equation.

When working on equations like \(6+\frac{1}{x}=\frac{7}{x}\), you initially want to isolate the terms involving \(x\). This can involve moving terms from one side to another by performing the opposite of the operations given.
Sometimes, solving an equation might reveal that no solution exists or that the given equation was based on incorrect premises. That's why it's crucial to verify your final answer.

Equations with Fractions

Equations with fractions can seem tricky at first, but they're manageable with a few simple steps. Fractions appear when terms in equations have numerators and denominators. Solving these requires carefully managing these fractional components.
Consider the equation \(6+\frac{1}{x}=\frac{7}{x}\) from the original problem. The fractions here involve the denominator \(x\).
Here are strategies to deal with fractions:

• Find a common denominator to deal with fractions on each side easily.
• Multiply through by this common denominator to eliminate fractions.
• Simplify the resulting equation to find the variable's value.

In many cases, multiplying both sides by the denominators can help eliminate fractional parts, transforming the equation into a simpler form. Always remember to keep an eye on restrictions that denominators impose, as they must never be zero. This might affect the possible solutions of your equation.