Problem 55 - Algebra and Trigonometry

Question

Solve: \(\frac{x+3}{4}-\frac{x+1}{10}=\frac{x-2}{5}-1\) (Section \(1.2, \text { Example } 3)\)

Step-by-Step Solution

Answer

The solution is \(x=-41\).

Step 1: Clear fractions

Multiply the entire equation by the least common multiple (LCM) of the denominators, which is 20 here. This gives us: \(5(x+3) - 2(x+1) = 4(x-2) - 20 \).

Step 2: Distribute

Distribute the numbers to the parentheses which gives us: \(5x+15 - 2x-2 = 4x-8 -20\).

Step 3: Simplify the equation

Combine like terms on each side of the equation: \(3x+13 = 4x-28\).

Step 4: Solve for x

Now isolate x on one side: Subtract 3x from both sides, results in \(x=-41\).

Key Concepts

Clearing Fractions in Equations

To make solving linear equations easier, a helpful initial step is to get rid of any fractions. Clearing fractions in an equation involves multiplying each term by a common number that allows you to have integers rather than fractional coefficients. The process simplifies the equation and reduces the chances for mistakes.

For the equation \(\frac{x+3}{4}-\frac{x+1}{10}=\frac{x-2}{5}-1\), the common number used to clear the fractions is the least common multiple of all the denominators (4, 10, and 5), which is 20. Multiplying every term by 20, we avoid working with fractions altogether, resulting in a clearer, more manageable equation.

Least Common Multiple

The least common multiple (LCM) of two or more numbers is the smallest number that's a multiple of all of them. It's essential when combining fractions to have the same denominators or when clearing fractions as in our example.

To find the LCM of 4, 10, and 5 which are the denominators in our initial equation, we list out the multiples of each number. The LCM is the first shared value that appears in all lists, which in this case is 20. This value is used to multiply each term in the equation to eliminate the fractional parts, setting up the equation for easier manipulation.

Combining Like Terms

Combining like terms is a fundamental aspect of simplifying algebraic expressions and equations. Like terms are terms that contain the same variables raised to the same power. The coefficients of these terms can be added or subtracted to combine them into a single term, making the equation simpler.

For instance, in the simplified equation after clearing fractions, \(5x+15 - 2x-2 = 4x-8 -20\), the like terms are those with the variable \(x\) and the constants. On the left side, \(5x\) and \( -2x\) can be combined, as can \(15\) and \( -2\). Following similar principles, the right side simplifies to \(4x-28\), preparing the equation for the final solution.

Distributive Property

The distributive property is a cornerstone of algebra which allows us to remove parentheses by distributing a multiplier to each term within the parentheses. Typically represented by the formula \(a(b+c) = ab + ac\), it enables us to simplify expressions and solve equations effectively.

In our example, after multiplying by the LCM, we use the distributive property to simplify terms. We take the multipliers outside the parentheses, \(5\), \( -2\), and \(4\), and distribute them to the terms inside, leading to the equation \(5x+15 - 2x-2 = 4x-8 -20\). This step is crucial because it provides a clearer path towards combining like terms and ultimately solving for \(x\).