Problem 19
Question
Write an equation that expresses each relationship. Then solve the equation for y. \(x\) varies directly as \(z\) and inversely as the difference between \(y\) and \(w\)
Step-by-Step Solution
Verified Answer
The equation that represents this relationship is \(y = \frac{c}{kz} + w\)
1Step 1: Establish the direct variation
Firstly, you need to establish the direct variation. If x varies directly as z, the equation representing this variation is \(x = kz\), where k is constant of variation.
2Step 2: Establish the inverse variation
Next, to establish the inverse variation. If x varies inversely as the difference between y and w, the given statement can be represented by the equation \(x = \frac{c}{y - w}\), where c is another constant.
3Step 3: Merge the two relationships
As it's given that x varies both directly as z and inversely as the difference of y and w, you can merge the two relationships in one equation, because they are both equal to x. So, \(kz = \frac{c}{y - w}\)
4Step 4: Solve the equation for y
To solve for y, the equation needs to be rearranged. This can be done by multiplying both sides by \(y - w\), to get \(kz(y - w) = c\). Then dividing both sides by \(kz\), to get \(y - w = \frac{c}{kz}\). Finally, adding w to both sides yields the solution \(y = \frac{c}{kz} + w\).
Key Concepts
Variation ConstantSolving EquationsAlgebraic Manipulation
Variation Constant
In the context of variation, a constant is a fixed value that describes the specific direct or inverse relationship between variables. These constants are crucial as they allow us to define the proportional aspect of the variables involved in the equation.
For direct variation, if a variable \( x \) varies directly as another variable \( z \), the relationship is given by the equation \( x = kz \), where \( k \) is the constant of variation. This \( k \) scales the value of \( z \) to equate it to \( x \).
Inversely, when a variable \( x \) varies inversely with another expression or variable, such as the difference \( y - w \), it means as one increases, the other decreases. It's represented by \( x = \frac{c}{y - w} \), where \( c \) is the constant of variation for the inverse relationship. Together, these constants help us understand and quantify how variables influence each other.
For direct variation, if a variable \( x \) varies directly as another variable \( z \), the relationship is given by the equation \( x = kz \), where \( k \) is the constant of variation. This \( k \) scales the value of \( z \) to equate it to \( x \).
Inversely, when a variable \( x \) varies inversely with another expression or variable, such as the difference \( y - w \), it means as one increases, the other decreases. It's represented by \( x = \frac{c}{y - w} \), where \( c \) is the constant of variation for the inverse relationship. Together, these constants help us understand and quantify how variables influence each other.
Solving Equations
Solving equations involves finding the values of unknown variables that satisfy the equation. It is a key skill in algebra and allows us to decode complex relationships between varying quantities.
When solving an equation like \( kz = \frac{c}{y - w} \), you first need to isolate the term involving the variable you are solving for. In this case, we want to solve for \( y \). Begin by performing algebraic manipulations such as multiplying both sides by \( y - w \), resulting in \( kz(y - w) = c \). This step eliminates any fractions, making the equation easier to work with.
By rearranging and simplifying the equation, you'd arrive at \( y - w = \frac{c}{kz} \). Finally, to completely solve for \( y \), add \( w \) to both sides, leaving \( y = \frac{c}{kz} + w \). This gives us the explicit expression for \( y \) in terms of the other variables.
When solving an equation like \( kz = \frac{c}{y - w} \), you first need to isolate the term involving the variable you are solving for. In this case, we want to solve for \( y \). Begin by performing algebraic manipulations such as multiplying both sides by \( y - w \), resulting in \( kz(y - w) = c \). This step eliminates any fractions, making the equation easier to work with.
By rearranging and simplifying the equation, you'd arrive at \( y - w = \frac{c}{kz} \). Finally, to completely solve for \( y \), add \( w \) to both sides, leaving \( y = \frac{c}{kz} + w \). This gives us the explicit expression for \( y \) in terms of the other variables.
Algebraic Manipulation
Algebraic manipulation is the process of rearranging algebraic expressions and equations by applying mathematical operations. It enables us to derive meaningful solutions from given relationships, often involving more than one variable.
Some common steps in algebraic manipulation include:
In the example where \( kx = \frac{c}{y - w} \), understanding which operations to use is key. Multiplying both sides by \( y - w \) and dividing by \( kz \) required precise knowledge of how to clear fractions and eventually allow \( y \) to be isolated. These manipulations are the backbone of solving algebraic equations effectively.
Some common steps in algebraic manipulation include:
- Adding, subtracting, multiplying, or dividing both sides of an equation to isolate the variable.
- Combining like terms to simplify expressions.
- Utilizing properties of equality for solving, such as the distributive property.
In the example where \( kx = \frac{c}{y - w} \), understanding which operations to use is key. Multiplying both sides by \( y - w \) and dividing by \( kz \) required precise knowledge of how to clear fractions and eventually allow \( y \) to be isolated. These manipulations are the backbone of solving algebraic equations effectively.
Other exercises in this chapter
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