Question

Solve and check each equation. Treat the constants in these equations as exact numbers. Leave your answers in fractional, rather than decimal, form. $$25 x-5=3 x+6$$

Short answer

The solution to the equation is \(x = \frac{1}{2}\).

Step-by-step solution

Isolate Variable

To solve for x, we first isolate the variable on one side by subtracting the variable term on the right side from both sides. We do this by subtracting 3x from both sides of the equation: $$25x - 3x -5 = 3x - 3x + 6$$.

Simplify Both Sides

Simplify both sides of the equation by combining like terms: $$22x - 5 = 6$$.

Move the Constant Term

Move the constant term on the left side of the equation to the other side by adding 5 to both sides of the equation: $$22x - 5 + 5 = 6 + 5$$.

Simplify the Equation

Simplify the equation by combining the constant terms: $$22x = 11$$.

Divide by the Coefficient of x

Divide both sides of the equation by the coefficient of x to solve for x: $$x = \frac{11}{22}$$.

Simplify the Fraction

Finally, simplify the fraction to find the value of x: $$x = \frac{1}{2}$$.

Checking the Solution

Substitute \(x = \frac{1}{2}\) back into the original equation to check the solution: $$25(\frac{1}{2}) - 5 = 3(\frac{1}{2}) + 6$$.Simplify both sides to verify they are equal: $$\frac{25}{2} - \frac{10}{2} = \frac{3}{2} + \frac{12}{2}$$which simplifies to$$\frac{15}{2} = \frac{15}{2}$$.Since both sides are equal, the solution is verified.

Key concepts

Isolating Variables

When solving linear equations, one of the first steps is often to isolate the variable. This means manipulating the equation so that the variable you're solving for is by itself on one side of the equality. To achieve this, you can perform operations such as addition, subtraction, multiplication, or division to both sides of the equation, ensuring you maintain the balance.

In the given problem, the first step is to subtract the term containing the variable on the right side from both sides. This action isolates the variable on the left side, setting the stage for subsequent simplifications. Always remember to perform the same operation on both sides to keep the equation balanced.

Combining Like Terms

Combining like terms is a critical step in simplifying an equation. Like terms are terms that have the same variables raised to the same power. In other words, they're parts of the equation that can be combined because they are similar.

In the example problem, after isolating the variable, we combine the like terms on each side to further simplify the equation. This step often involves adding or subtracting coefficients (the numerical parts of the terms) and can significantly reduce the complexity of the problem. Understanding how to effectively combine like terms makes solving equations much more manageable.

Checking Equations Solutions

After finding a solution to an equation, it is just as important to check that the solution is correct. This verification step ensures that the operations performed have led to a valid solution.

To check the solution of the exercise, the found value for the variable is substituted back into the original equation. If both sides of the equation evaluate to the same numerical value after the substitution, it confirms that the solution is correct. In this case, substituting \(x = \frac{1}{2}\) back into the equation confirmed that the solution was indeed correct since both sides simplify to the same value, \(\frac{15}{2}\).

Solving Fractional Equations

Fractional equations often intimidate students, but they can be solved using the same principles as other linear equations. The primary goal is to eliminate the fractions to simplify the solving process.

However, in our exercise, after isolating the variable and combining like terms, we ended up with a simple fraction as the solution. Simplifying the fraction is the last step. In this instance, when we divided both sides by the coefficient of the variable (22), we obtained \(\frac{11}{22}\), which simplifies to \(\frac{1}{2}\). When dealing with fractional equations, always look for opportunities to simplify fractions to their lowest terms to achieve the most straightforward form of the solution.