Problem 4
Question
The ________ products for the proportion \(\frac{10}{3}=\frac{5}{x}\) are \(10 x\) and 15
Step-by-Step Solution
Verified Answer
The cross products are \(10x\) and 15.
1Step 1: Identify the Equation
We are given the proportion equation: \( \frac{10}{3} = \frac{5}{x} \). In proportions, cross-multiplying can be used to solve for the unknown variable.
2Step 2: Cross-Multiply
To find the cross products, multiply the numerator of one fraction by the denominator of the other fraction: \( 10 \times x \) and \( 3 \times 5 \).
3Step 3: Write Out Cross Products
The cross products for the given proportion are \( 10x \) and \( 15 \).
4Step 4: Equate the Cross Products
In a proportion, the cross products are equal: \( 10x = 15 \).
5Step 5: Solve for x
Divide both sides of the equation \( 10x = 15 \) by 10 to isolate \( x \): \( x = \frac{15}{10} = 1.5 \).
Key Concepts
Understanding Cross-MultiplicationBasics of Solving EquationsDemystifying the Unknown Variable
Understanding Cross-Multiplication
Cross-multiplication is a powerful technique used in solving proportions. A proportion is an equation that states two ratios are equal, such as \( \frac{10}{3} = \frac{5}{x} \). Cross-multiplying involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the results equal to each other. This process helps us clear the fractions and make the problem easier to solve.
For example, in the proportion \( \frac{10}{3} = \frac{5}{x} \), you multiply 10 by \( x \) and 3 by 5, resulting in two products: \( 10 \times x \) and \( 3 \times 5 \), which gives us \( 10x \) and 15. These are known as the cross products. Cross-multiplication helps us set up an equation without fractions: \( 10x = 15 \). This technique is particularly helpful because it simplifies the problem by turning it into a basic algebraic equation. It's a neat trick that makes it much easier to tackle proportions!
For example, in the proportion \( \frac{10}{3} = \frac{5}{x} \), you multiply 10 by \( x \) and 3 by 5, resulting in two products: \( 10 \times x \) and \( 3 \times 5 \), which gives us \( 10x \) and 15. These are known as the cross products. Cross-multiplication helps us set up an equation without fractions: \( 10x = 15 \). This technique is particularly helpful because it simplifies the problem by turning it into a basic algebraic equation. It's a neat trick that makes it much easier to tackle proportions!
Basics of Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. Once you've set up the equation from a proportion with cross-multiplication, the next step is to solve it using basic algebraic principles.
Take our equation: \( 10x = 15 \). The goal is to isolate the variable, which in this case is \( x \). To do that, you need to perform operations that will give you \( x \) on one side of the equation and a number on the other. Here, you'll want to divide both sides of the equation by 10 to get \( x \) by itself.
Here’s how it looks:
Take our equation: \( 10x = 15 \). The goal is to isolate the variable, which in this case is \( x \). To do that, you need to perform operations that will give you \( x \) on one side of the equation and a number on the other. Here, you'll want to divide both sides of the equation by 10 to get \( x \) by itself.
Here’s how it looks:
- Divide each side by 10: \( \frac{10x}{10} = \frac{15}{10} \)
- This simplifies to \( x = 1.5 \)
Demystifying the Unknown Variable
The mystery of the unknown variable often seems daunting to students at first, but it's simpler than it looks. In our example, the unknown variable is \( x \), and our task is to find its value using algebraic methods.
An unknown variable in an equation represents a quantity that is hidden or not yet determined. It's a placeholder that will be replaced by a number once the equation is solved. In our proportion \( \frac{10}{3} = \frac{5}{x} \), \( x \) signifies the unknown that we aim to uncover. By setting up the equation and using techniques like cross-multiplication, we can reveal what \( x \) truly is.
Through the process of solving, the unknown is clarified, leading us to discover that \( x = 1.5 \). This sheds light on the fact that the variable can be any value that satisfies the equation. Understanding that the unknown is simply a number waiting to be discovered demystifies the process and makes solving equations less intimidating. Remember, the variable isn't there to trick you; it's just waiting to be found!
An unknown variable in an equation represents a quantity that is hidden or not yet determined. It's a placeholder that will be replaced by a number once the equation is solved. In our proportion \( \frac{10}{3} = \frac{5}{x} \), \( x \) signifies the unknown that we aim to uncover. By setting up the equation and using techniques like cross-multiplication, we can reveal what \( x \) truly is.
Through the process of solving, the unknown is clarified, leading us to discover that \( x = 1.5 \). This sheds light on the fact that the variable can be any value that satisfies the equation. Understanding that the unknown is simply a number waiting to be discovered demystifies the process and makes solving equations less intimidating. Remember, the variable isn't there to trick you; it's just waiting to be found!
Other exercises in this chapter
Problem 4
Fill in the blanks. By the ____ theorem, if a polynomial \(P(x)\) is divided by \(x-k,\) the remainder is \(P(k)\).
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Check to determine whether 2 is a solution of the following equations. a. \(\frac{x}{2}+\frac{4}{x+2}=x\) b. \(\frac{x+2}{x-2}+\frac{1}{x^{2}-4}=1\)
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Fill in the blanks. The polynomials \(x-y\) and \(y-x\) are ______ because their terms are the same but opposite in sign.
View solution