Question

Solve the equation. Check your solution. $$ \frac{1}{4} x-\frac{3}{5}=\frac{9}{2} x-\frac{4}{5} $$

Short answer

The solution for the equation is \( x = - \frac{4}{85} \).

Step-by-step solution

Simplify the Equation

To make it easier, the first step is to multiply each term by the least common multiple (LCM) of the denominators (4,5,2), which is 20. So, the equation \( \frac{1}{4}x - \frac{3}{5} = \frac{9}{2}x - \frac{4}{5} \) becomes \( 20* \frac{1}{4}x -20* \frac{3}{5} = 20* \frac{9}{2}x - 20*\frac{4}{5} \) Which simplifies to \( 5x - 12 = 90x - 16 \).

Grouping Like Terms

Next, we isolate x terms on one side of the equation and the constant terms on the other side. Thus we write the equation as \( 5x - 90x = 16 - 12 \) Which simplifies to \( -85x = 4 \).

Solving for the Variable

Finally, we solve for x by dividing the equation through by -85, to get \( x = \frac{4}{-85} \) is equal\( x = - \frac{4}{85} \).

Checking the Solution

We substitute \( x = - \frac{4}{85} \) back into the original equation \( \frac{1}{4}x - \frac{3}{5} = \frac{9}{2}x - \frac{4}{5} \) . After substitution we get \( \frac{1}{4}*(-\frac{4}{85}) - \frac{3}{5} = \frac{9}{2}*(-\frac{4}{85}) - \frac{4}{5} \) Simplifying on both sides confirms that left side is equal to the right, hence the value of x obtained is correct.

Key concepts

Least Common Multiple

When you're solving linear equations with fractions, it's important to first look at the denominators of the fractions. Here, we aim to simplify the equation by removing these denominators. The tool we use for this task is finding the Least Common Multiple (LCM).
To do this, you list the multiples of each denominator and find the smallest number common to them all. For denominators 4, 5, and 2, the LCM is 20.
By multiplying every term of the equation by the LCM, you eliminate the fractions, making it simpler to solve. This step should always be your starting point when dealing with fractions in equations.

Simplifying Expressions

Once you've multiplied through by the LCM, your equation should no longer contain fractions. The next step is to simplify the expressions on both sides of the equation. This involves distributing the multiplication across terms and combining any like terms.
In our example, after multiplying each term by 20, you simplify it to be free of fractions, bringing the equation to a form like \(5x - 12 = 90x - 16\).

• Multiply and distribute the LCM across terms in the equation.
• Combine like terms if applicable.
• Check that each side of the equation is fully simplified.

Simplifying correctly helps make the next steps of solving the equation easier.

Substitution Method

When checking your solution, you'll often use the substitution method. This involves plugging the solution back into the original equation to ensure it holds true.
For our equation, upon substituting \(x = -\frac{4}{85}\) back into the original equation, ensure you calculate each side separately.
Reworking the left and right sides confirms whether both yield the same value, verifying the solution's correctness. This process reassures your solution is accurate.

• Substitute the solution back into the original equation.
• Simplify both sides and compare.
• Ensure that both sides are equal.

The substitution method is a crucial step in verifying solutions.

Isolating Variables

Isolating the variable is a key step in solving any equation. After simplification, you want all terms containing your variable on one side and constant terms on the other.
In our example, bring all terms involving \(x\) on one side and constant terms on the opposite side. So from \(5x - 90x = 16 - 12\), which simplifies further to \(-85x = 4\).
Then, solve for \(x\) by dividing both sides by \(-85\). This gives \(x = -\frac{4}{85}\).

• Move terms to isolate the variable on one side.
• Use inverse operations, like division, to solve for the variable.
• Check your equation is in the form variable = constant.

Mastering this skill makes solving linear equations straightforward and methodical.