Problem 27 - Algebra 1: Concepts and Skills

Question

Solve the equation. Check your solutions. $$ \frac{x-2}{4}=\frac{x+10}{10} $$

Step-by-Step Solution

Answer

The solution to the equation is \(x=10\).

Step 1: Clear out the fractions

The fractions can be eliminated by multiplying both sides of the equation by the lowest common multiple (LCM) of the denominators 4 and 10, which is 20. This gives: \(20*(x-2)/4 = 20*(x+10)/10\) This simplifies to: \(5x-10= 2x+20\)

Step 2: Simplify and solve for x

Subtract 2x from both sides to get all x terms on one side: \(5x-2x = 10+20\) or \(3x=30\). Then divide by 3 to isolate x: \(x=30/3\) or \(x=10\)

Step 3: Check the solution by substitution

Substitute 10 for x in the original equation: \( (10-2)/4 = (10+10)/10 \) This simplifies to: \( 2=2 \), indicating that x=10 is a valid solution to the given equation.

Key Concepts

Fractions in Equations

When working with equations containing fractions, many students can initially find them intimidating. However, the key is to understand that fractions are simply another way of expressing division. The aim when dealing with these types of equations is to eliminate the fractions to more easily isolate the variable.

To clear out the fractions in an equation like \( \frac{x-2}{4}=\frac{x+10}{10} \), it is beneficial to find a common denominator or better yet, use the least common multiple (LCM) of the denominators. For example, the LCM of 4 and 10 is 20. By multiplying every term by this LCM, you effectively cancel out the denominators.

• This step converts each term into an integer, simplifying the equation considerably.
• After this transformation, working with the equation becomes much simpler, without needing to worry about fractions.

By carefully handling fractional terms, one can solve linear equations more confidently and accurately.

Simplifying Equations

Once the fractions are eliminated, the next task is simplifying the expression to reveal the solution. Simplifying is about reducing equations to their most basic form so the solution can be seen more clearly.

In the original equation \( \frac{x-2}{4} = \frac{x+10}{10} \), once multiplied out and fractions are cleared, it transforms into \( 5x - 10 = 2x + 20 \). Here are steps to continue simplifying:

• Bring like terms together, in this case, focus on \( x \) terms first. Reduce the equation by subtracting \( 2x \) from both sides.
• This leaves you with \( 3x = 30 \).

Now, solve for \( x \) by dividing each side of the equation by 3, resulting in \( x = 10 \).

Simplifying through such steps reduces complexity and makes finding the solution straightforward.

Checking Solutions

Once a solution is found, it's important to check that it truly satisfies the original equation. Returning to the original equation ensures accuracy in problem-solving.

For the solution \( x = 10 \), substitute back into the original equation \( \frac{x-2}{4}=\frac{x+10}{10} \):

• Substitute \( x = 10 \) to get \( \frac{10-2}{4} = \frac{10+10}{10} \).
• Simplify both sides: \( 2 = 2 \).

Both sides of the equation are equal, confirming that \( x = 10 \) is indeed a valid solution.

Checking your answers is vital because it verifies the accuracy of the solution. It is an indispensable step in solving mathematical problems confidently and accurately.