Problem 38 - Elementary Algebra

Question

Solve the equations. $$ \frac{6 x-1}{7}=-3 $$

Step-by-Step Solution

Answer

Answer: $x = -\frac{10}{3}$

Step 1: Multiply by 7

To eliminate the fraction, we can multiply both sides of the equation by 7: $$ 7\cdot\frac{6 x-1}{7} = (-3)\cdot 7 $$ This gives: $$ 6x - 1 = -21 $$

Step 2: Simplify and collect terms

Now, let's simplify the equation and collect the terms: $$ 6x - 1 + 1 = -21 + 1 $$ This simplifies to: $$ 6x = -20 $$

Step 3: Solve for x

To find the value of x, we can divide both sides of the equation by 6: $$ x = \frac{-20}{6} $$ Simplifying the fraction, we find the solution: $$ x = -\frac{10}{3} $$

Key Concepts

Fraction Elimination

Fractions can often make equations look more complicated, but there's a straightforward trick to simplify them. This is called fraction elimination. When you have a fraction in an equation, you can eliminate it by multiplying both sides by the denominator of the fraction. This works because multiplying by the denominator effectively "cancels out" the fraction, leaving you with a simplified equation to work with.

• For example, if your equation is \(\frac{a}{b} = c\), multiplying both sides by \(b\) will give you \(a = bc\).
• This step is crucial as it transforms a complex-looking problem into a much simpler one.
• By understanding fraction elimination, you gain a powerful tool to handle many types of equations, quickly reducing them to a more manageable form.

In our exercise, by multiplying both sides of \(\frac{6x - 1}{7} = -3\) by 7, the fraction disappears, leaving \(6x - 1 = -21\), an equation without fractions.

Linear Equations

Linear equations are equations where the highest power of the variable is one. They are called 'linear' because they graph as straight lines. These equations usually look like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants. Solving these equations involves finding the value of the variable that makes the equation true.

• The main steps in solving linear equations include isolating the variable on one side of the equation.
• This typically involves moving terms from one side to the other through addition or subtraction, and then dividing or multiplying to solve for the variable.

In our example, after eliminating the fraction, we have \(6x - 1 = -21\). We then add 1 to both sides to isolate the term with the variable, resulting in \(6x = -20\). This brings us closer to solving for \(x\).

Simplifying Expressions

Simplifying expressions involves reducing them to their most basic form while maintaining the equation's integrity. This means performing any possible arithmetic and reducing fractions if necessary.
When you simplify, you often:

• Combine like terms to make the expression shorter.
• Reduce fractions to their simplest forms.

For example, after isolating \(6x = -20\), we divide both sides by 6. The right side simplifies to \(x = \frac{-20}{6}\). Next, to express this fraction in its simplest form, we divide both the numerator and the denominator by their greatest common divisor, which is 2, resulting in \(x = -\frac{10}{3}\).
Simplifying expressions helps to visualize the equation's solution and ensures your answer is presented in the clearest, universally understood form.