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Chapter 2: Problem 24

For each formula, express y as a function of x. $$\frac{1}{3} x-\frac{1}{4} y=1$$

Short Answer

Expert verified
y = \frac{4}{3} x - 4

Step by step solution

01

Isolate the term involving y

Start by isolating the term that includes y. Move the constant term to the other side of the equation. Begin with the original equation: \[ \frac{1}{3} x - \frac{1}{4} y = 1 \] Next, subtract \( \frac{1}{3} x \) from both sides: \[ - \frac{1}{4} y = 1 - \frac{1}{3} x \]
02

Eliminate the coefficient of y

Remove the coefficient of y by multiplying both sides of the equation by -4. \[ y = -4 \left( 1 - \frac{1}{3} x \right) \] Simplify inside the parentheses: \[ y = -4 + \left( \frac{4}{3} x \right) \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables is a crucial step in solving algebraic equations. It involves rearranging the equation to get the variable of interest on one side. This makes it easier to solve for that variable. In our example, we need to express y as a function of x from the equation: We start by isolating the term that involves y. Move the constant term to the other side. Subtract from both sides to isolate the term with y: This step sets the stage for solving the equation, making sure the variable y is separated on one side of the equation.
Solving Linear Equations
Solving linear equations often requires a few steps to isolate the variable and solve for it. A linear equation is an equation of the form . In our case, we start with . We have already isolated the term with y in the preceding step. Now, we need to eliminate the coefficient of y by performing operations that simplify the equation. Next, we eliminate the coefficient of y. In this example, the coefficient is . To remove it, multiply both sides of the equation by -4: This multiplication allows us to cancel out the coefficient, leaving y alone on one side of the equation. We then simplify further:
Coefficients
Understanding coefficients is essential when solving algebraic equations. A coefficient is the numerical factor of a term that includes a variable. In the equation The coefficients are for x and for y. Knowing how to work with coefficients is key to simplifying and solving equations. To solve for y, you must deal with its coefficient, which is - . By multiplying both sides by -4, we effectively 'cancel out' the fraction, making it easier to isolate y.The final step involves simplifying inside the parentheses: Remember, simplifying the expression ensures that we have y completely isolated in terms of x.

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Most popular questions from this chapter

Show a complete solution to each problem. Mary's assets. Mary Hall's will specifies that her lawyer is to liquidate her assets and divide the proceeds among her three sisters. Lena's share is to be one-half of Lisa's, and Lisa's share is to be one-half of Lauren's. If the lawyer has agreed to a fee that is equal to \(10 \%\) of the largest share and the proceeds amount to \(\$ 164,428,\) then how much does each person get?

A pair of ordinary jeans at A-Mart costs \(\$ 50\) less than a pair of designer jeans at Enrico's. In fact, you can buy four pairs of A-Mart jeans for less than one pair of Enrico's jeans. What is the price range for a pair of A-Mart jeans?

Solve problem by using an inequality. Car shopping. Jennifer is shopping for a new car. In addition to the price of the car, there is an \(8 \%\) sales tax and a \(\$ 172\) title and license fee. If Jennifer decides that she will spend less than \(\$ 10,000\) total, then what is the price range for the car?

Solve each equation. Practice combining some steps. Look for more efficient ways to solve each equation. $$3 y+5=4 y-1$$

Explain why saying that \(x\) is at least 9 is equivalent to saying that \(x\) is greater than or equal to \(9 .\) Explain why saying that \(x\) is at most 5 is equivalent to saying that \(x\) is \(=\) less than or equal to \(5 .\)
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