Question

Find the zeros of the function \(\\{(2 \mathrm{x}+7) / 5\\}+\\{(3 \mathrm{x}-5) / 4\\}+(33 / 10)\)

Short answer

The zeros of the function \(\frac{2x+7}{5} + \frac{3x-5}{4} + \frac{33}{10}\) can be found by simplifying the function and solving for x when the function equals 0. The simplified function is \(\frac{23x + 69}{20}\). Setting the simplified function equal to 0, we find that the zero of the function is \(x = -3\).

Step-by-step solution

Write down the given function

The given function is \[\frac{2x+7}{5} + \frac{3x-5}{4} + \frac{33}{10}\].

Simplify the function

To simplify the function, first, we will find the least common denominator (LCD) for the three expressions. We have denominators 5, 4, and 10. The LCD for these denominators is 20. Now, we will rewrite the fractions using the LCD. \[\frac{2x+7}{5} \times \frac{4}{4} + \frac{3x-5}{4} \times \frac{5}{5} + \frac{33}{10} \times \frac{2}{2} = \frac{8x+28}{20} + \frac{15x-25}{20} + \frac{66}{20}\] Now, we will sum the three fractions and simplify the resulting expression. \[\frac{8x+28 + 15x-25 + 66}{20} = \frac{23x + 69}{20}\]

Write the equation with the simplified function equal to 0

We need to find the value(s) of x for which the function equals 0. So, we have: \[\frac{23x + 69}{20}=0\]

Solve for x

To solve for x, we will multiply both sides of the equation by the denominator (20), so that we can get rid of the fractions. \[20 \times \frac{23x + 69}{20}=0 \times 20\] \[23x + 69 = 0\] Now, we will subtract 69 from both sides of the equation to isolate x. \[23x = -69\] Finally, divide by 23 to find the value of x. \[x = \frac{-69}{23}\] So, the zero of the given function is \(x = -3\).

Key concepts

Simplifying Expressions

When working with algebraic expressions, simplifying them makes it easier to understand and solve equations. Simplification generally involves reducing an expression to its most basic form without changing its value. This can include expanding products, combining like terms, and canceling common factors. For example, in our exercise, the expression \(\frac{2x+7}{5} + \frac{3x-5}{4} + \frac{33}{10}\) was simplified by finding a common denominator for all terms.

To simplify an expression effectively, we follow specific algebraic rules and processes. These can include distributive properties, associative and commutative properties, and arithmetic operations. With these rules, we enhance the clarity of the expression, allowing us to see the core components that we'll work with when solving equations.

Lowest Common Denominator

Finding the lowest common denominator (LCD) is key when dealing with fractions, especially when simplifying or adding them. The LCD is the smallest number that all denominators in a set of fractions can divide into evenly. For instance, if you have fractions with denominators 5, 4, and 10, like in our exercise, you'd need a number that all three of these numbers go into without leaving a remainder. In this case, that's 20.

Using the LCD to rewrite expressions ensures all fractions have the same denominator, making it simple to combine them or compare their sizes. It's like getting everybody to speak the same language before starting a discussion - it sets a common ground for all the terms involved. To find the LCD, one often uses the least common multiple (LCM) of the denominators or multiples of the highest power of prime factors present in the denominators.

Solving Equations

Solving equations is a fundamental skill in algebra. It's all about isolating the variable, which means getting the variable on one side of the equation and everything else on the other. In the exercise, our mission was to find the zeros of a function, which means finding the value of \(x\) that makes the function equal to zero. To do this, we started by simplifying the expression and then rewrote the simplified function as an equation set to zero, \(\frac{23x + 69}{20}=0\).

Then, by multiplying both sides by the common denominator, we eliminated the fraction. From there, it was a matter of isolating \(x\) by using inverse operations - subtracting 69 and then dividing by 23 - yielding the zero of the function. The process of solving equations often involves reverse operations of basic arithmetic and understanding how to manipulate expressions while maintaining equality.