Question

If $\frac{2}{17}x=15$, what is $\frac{6}{17}x$ ? (Note: You may not use a calculator.)

Short answer

The value of $\frac{6}{17}x$ is $\frac{255}{17}$.

Step-by-step solution

Find the value of x from the given equation

We are given the equation $\frac{2}{17}x=15$. To find the value of $x$, we need to isolate $x$ by multiplying both sides of the equation by the inverse of the coefficient of $x$, which is $\frac{17}{2}$: $\frac{17}{2}\times \frac{2}{17}x=15\times \frac{17}{2}$ The $\frac{17}{2}$ and $\frac{2}{17}$ will cancel each other out on the left side of the equation: $x=15\times \frac{17}{2}$

Calculate the value of x

Now, we can simplify the right side of the equation to find the value of $x$: $x=15\times \frac{17}{2}$ $x=\frac{15\times 17}{2}$ $x=\frac{255}{2}$ So, the value of $x$ is $\frac{255}{2}$.

Find the value of $\frac{6}{17}x$

Now that we have the value of $x$, we can plug it into the expression $\frac{6}{17}x$ to find its value: $\frac{6}{17}x=\frac{6}{17}\times \frac{255}{2}$

Simplify the expression

Now, we can simplify the expression to find the value of $\frac{6}{17}x$: $\frac{6}{17}\times \frac{255}{2}=\frac{6\times 255}{17\times 2}$ We can simplify further by cancelling out common factors: $\frac{3\times 85}{17\times 1}$ $\frac{3\times 85}{17}=\frac{255}{17}$ So, the value of $\frac{6}{17}x$ is $\frac{255}{17}$.

Key concepts

Equation solving

In algebra, solving equations is about finding the unknown value that makes an equation true. An equation is a mathematical statement that two expressions are equal. To solve an equation, we aim to isolate the variable on one side of the equation. This process often involves undoing operations using inverse operations. Consider the equation $\frac{2}{17}x=15$. The goal is to solve for the unknown variable $x$. Here, $x$ is being multiplied by the fraction $\frac{2}{17}$. To isolate $x$, we multiply both sides of the equation by the reciprocal of $\frac{2}{17}$, which is $\frac{17}{2}$.

• Multiplying by the reciprocal cancels out the fraction on the left side, leaving $x$ by itself.
• On the right side, perform the multiplication to get $x=15\times \frac{17}{2}$.

This method ensures that the equation remains balanced and $x$ is correctly isolated.

Fraction operations

Working with fractions in algebra involves understanding how to multiply, divide, add, or subtract them effectively. Fraction operations follow certain rules, making it important to know techniques like finding a common denominator or simplifying fractions.Here, we focus on multiplying fractions as seen in the initial equation $\frac{2}{17}x=15$. When handling fractions:

• Multiply the numerators together and the denominators together. For example, $\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}$.
• After you multiply, you can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD).

This problem involves multiplying $\frac{6}{17}$ by $x=\frac{255}{2}$. We execute the multiplication like this: $\frac{6\times 255}{17\times 2}$ and simplify by cancelling out the $3$ and $1$ in both the numerator and denominator, resulting in $\frac{255}{17}$. Understanding these fraction operations is key to manipulating and solving algebraic expressions effectively.

Mathematical expressions

In algebra, a mathematical expression is a combination of numbers, variables, and operators (such as +, -, ×, ÷) that represent a specific value or set of values. Expressions do not have an equality sign like equations do.To work with expressions like $\frac{6}{17}x$, it's important to interpret what the expression means and how changes to the variables or numbers affect its value.

• Substitute the known value of the variable back into the expression to find its new value.
• Simplify the expression by performing arithmetic operations as needed.

The expression $\frac{6}{17}x$ was evaluated by substituting $x=\frac{255}{2}$ into it. We then simplified $\frac{6\times 255}{17\times 2}$ to get $\frac{255}{17}$. By breaking down expressions in pieces, students can see their structure and how they work, providing greater insight into the relationships between numbers and variables in algebra.