Question

If \(y=2 / 3 x+5 / 6 x\) and \(x=1\) then \(y\) is: (A) \(2 / 3\) (B) \(7 / 9\) (C) \(7 / 3\) (D) \(7 / 6\) (E) None of the above

Short answer

The answer is none of the above because \(y=\frac{3}{2}\).

Step-by-step solution

Substitute the value of x

Start by substituting the given value of \( x \), which is \( x = 1 \), into the expression for \( y \). So, \( y = \frac{2}{3} x + \frac{5}{6} x \) becomes \( y = \frac{2}{3} (1) + \frac{5}{6} (1) \).

Simplify the expression

Simplify the expression by performing the multiplications: \( y = \frac{2}{3} + \frac{5}{6} \).

Find a common denominator

To add \( \frac{2}{3} \) and \( \frac{5}{6} \), find a common denominator. The least common denominator of 3 and 6 is 6. So, \( \frac{2}{3} \) can be rewritten as \( \frac{4}{6} \).

Add the fractions

Now add the fractions with the common denominator: \( \frac{4}{6} + \frac{5}{6} = \frac{9}{6} \).

Simplify the result

Simplify \( \frac{9}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3: \( \frac{9}{6} = \frac{3}{2} \).

Convert to mixed number (if necessary)

If necessary, convert \( \frac{3}{2} \) to a mixed number: \( 1 \frac{1}{2} \) (this step is just for understanding; it's not required here).

Key concepts

substitution method

The substitution method is a powerful tool you can use to solve equations easily. This method involves replacing one variable with its given value.
In our exercise, we are given that \( x = 1 \) and we need to find the value of \( y \).
Start by substituting \( x = 1 \) into the given equation: \( y = \frac{2}{3} x + \frac{5}{6} x \).
When you substitute \( x = 1 \) into the equation, it becomes \( y = \frac{2}{3} (1) + \frac{5}{6} (1) \). This simplifies some of the work for you.
By substituting, you're essentially making the problem smaller and more manageable.
It's like replacing a complicated part of a puzzle with an already known piece!

fractions simplification

Simplifying fractions is about making a fraction as simple as possible. You can do this by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Suppose our fraction is \( \frac{9}{6} \). First, find the GCD of 9 and 6, which is 3.
Now, divide both the numerator and the denominator by 3: \( \frac{9}{3} = 3 \) and \( \frac{6}{3} = 2 \).
This gives us a simplified fraction: \( \frac{9}{6} = \frac{3}{2} \).
Simplifying fractions not only makes the answer more readable but also easier to work with in further calculations.
For instance, rather than dealing with \( \frac{9}{6} \), you can use the simpler \( \frac{3}{2} \). It’s more straightforward and elegant.

common denominator

Finding a common denominator is essential when adding or subtracting fractions.
The common denominator is the smallest number that both denominators can divide into evenly.
In our problem, we need to add \( \frac{2}{3} \) and \( \frac{5}{6} \). The denominators here are 3 and 6.
A common multiple of 3 and 6 is 6. Therefore, 6 is our common denominator.
To add the fractions, convert \( \frac{2}{3} \) to a denominator of 6: \( \frac{2}{3} = \frac{4}{6} \).
Now, you can easily add: \( \frac{4}{6} + \frac{5}{6} = \frac{9}{6} \).
With a common denominator, adding fractions becomes much simpler, enabling seamless calculations!