Question

Solve for \(x\). Assume the integers in these equations to be exact numbers, and leave your answers in fractional form. \(\frac{x}{4}+\frac{x}{10}+\frac{x}{8}=19\)

Short answer

x = 40

Step-by-step solution

Find a common denominator

The fractions have denominators 4, 10, and 8. To combine them, we find a common denominator, which is 40 for these three numbers. Multiply each term by 40/40 to clear the denominators.

Clear the fractions

By multiplying each term by 40, the equation becomes: 10x + 4x + 5x = 760.

Combine like terms

Add the coefficients of x together: 10x + 4x + 5x = 19x.

Solve for x

Divide both sides of the equation by 19 to solve for x: x = 760 / 19.

Simplify the fraction

Divide 760 by 19 to simplify the fraction: x = 40.

Key concepts

Common Denominator

When solving equations involving fractions, it's often necessary to start by finding a common denominator. This is a shared multiple of the denominators in each term that allows us to add or subtract fractional expressions.

A common denominator can be thought of as a common ground, like a meeting point, for all the fractions in an equation. The lowest common denominator (LCD) is the smallest number that is a multiple of all the original denominators. In our example, the denominators are 4, 10, and 8. To find the LCD, we can list the multiples of each denominator and identify the smallest number that appears in all three lists. In this case, it’s 40.

Finding the common denominator is crucial because it simplifies the process of combining fractions. It's like bringing different time zone times into one standard time to make scheduling a global meeting much easier.

Clearing Fractions

Once a common denominator is established, the next step is clearing fractions from the equation. The goal here is to transform the equation into one without fractions, which often makes it much easier to solve. We achieve this by multiplying every term of the equation by the common denominator.

In our example, after finding that 40 is the common denominator, we multiply each term by 40/40, essentially multiplying by one, which does not change the value of the expression but does clear the fractions. The equation \(\frac{x}{4} + \frac{x}{10} + \frac{x}{8} = 19\) becomes 10x + 4x + 5x = 760 when multiplied by 40. This technique is a great way to simplify the process and proceed to the next steps with an equation that is free of fractions.

Combining Like Terms

With fractions out of the way, the next logical step is combining like terms. Like terms are terms in an equation that have the same variables raised to the same power. By combining them, we simplify the equation further.

After clearing the fractions in our example, we are left with 10x, 4x, and 5x on one side of the equation. Since they are all like terms (all terms are 'x' terms), we can combine them by adding the coefficients together. From 10x + 4x + 5x, we get a single term, 19x. This consolidation of terms is a bit like packing for a trip; instead of multiple small bags, we pack everything into one big suitcase for efficiency.

Simplifying Fractions

The final step in solving linear equations with fractions is often simplifying the fraction. Once you’ve determined the value for the variable, in this case, x, it’s important to express it in its simplest form.

After combining like terms, we find that the value of x is given by the equation x = 760 / 19. To simplify, we divide 760 by 19, which gives us x = 40. Simplifying fractions is like reducing a recipe to its most basic steps; it makes the result easier to work with and understand. This step confirms that the solution is in its most reduced form, which is often required in mathematical answers.