Question

Use the following information. In a direct variation, the ratio \(\frac{y}{x}\) is constant. If \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are solutions of the equation \(\frac{y}{x}=k,\) then \(\frac{y_{1}}{x_{1}}=k\) and \(\frac{y_{2}}{x_{2}}=k .\) Use the proportion \(\frac{y_{1}}{x_{1}}=\frac{y_{2}}{x_{2}}\) to find the missing value. Find \(x_{2}\) when \(x_{1}=2, y_{1}=3,\) and \(y_{2}=6\)

Short answer

The missing value of \(x_{2}\) is 4.

Step-by-step solution

Write down the given values

The values given in the exercise are \(x_{1}=2, y_{1}=3,\) and \(y_{2}=6\). We need to find \(x_{2}\).

Express the given values using the proportion

We know that \(\frac{y_{1}}{x_{1}}=\frac{y_{2}}{x_{2}}\). Plugging in the values, we get \(\frac{3}{2}=\frac{6}{x_{2}}\).

Solve for \(x_{2}\)

To obtain \(x_{2}\), we can cross-multiply and solve the equation \(\frac{3}{2}=\frac{6}{x_{2}}\). This simplifies to \(3x_{2}=12\). Solving for \(x_{2}\) gives \(x_{2}=4\).

Key concepts

Proportion

Understanding the concept of a proportion is crucial in many areas of mathematics, particularly when dealing with direct variations. A proportion is an equation that states that two ratios are equal. For instance, if we have two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), and we say \(\frac{a}{b} = \frac{c}{d}\), we're asserting that these two fractions are proportional to each other.

In real-life terms, proportions can describe relationships like the constant speed of a car (distance over time), ingredients in a recipe (amount of one ingredient over another), or in our textbook exercise, the relationship between two variables in direct variation. Whenever we see a proportion like \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\), we can conclude that \(y_1/x_1\) and \(y_2/x_2\) will always yield the same value, which is a foundational idea in solving problems involving direct variation.

Solving Proportions

Solving proportions is about finding the value of the unknown variable that makes the proportion true. One common method to solve proportions is cross-multiplication. This method involves multiplying the numerator of one ratio by the denominator of the other ratio and setting the two products equal to each other. For example, if we have a proportion like \(\frac{a}{b} = \frac{c}{d}\), we can find the unknown by cross-multiplying to get \(a \cdot d = b \cdot c\).

In our textbook example, we employed cross-multiplication to find \(x_2\) when the proportion \(\frac{3}{2} = \frac{6}{x_2}\) was given. By cross-multiplying, we obtained the equation \(3 \cdot x_2 = 2 \cdot 6\), leading to \(3x_2 = 12\), and after dividing by 3, we concluded \(x_2 = 4\). It's essential for learners to get comfortable with this method, as it's a powerful tool for solving a wide range of problems involving ratios and proportions.

Algebraic Equations

Algebraic equations are the bread and butter of algebra and are used to find unknown values that satisfy certain conditions. They involve constants and variables and use operations such as addition, subtraction, multiplication, and division. The basic goal when solving an algebraic equation is to isolate the variable, usually on one side of the equal sign, to determine its value.

The process of solving an equation can include simplifying expressions, combining like terms, and using inverse operations. For instance, if you have an equation like \(3x_2 = 12\), you would divide both sides by 3 to isolate the variable \(x_2\), arriving at \(x_2 = 4\). Mastering the manipulation of algebraic equations is vital for students as it forms a foundational skill set required for more advanced mathematics and various real-world applications.