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Chapter 2: Problem 15

Solve each equation for \(y.\) $$ 4 x+9 y=11 $$

Short Answer

Expert verified
y = \frac{11 - 4x}{9}

Step by step solution

01

Isolate the term with y

Start by moving the term with x to the other side of the equation. Subtract 4x from both sides to isolate the term with y: \[ 4x + 9y = 11 \] Subtract 4x from both sides: \[ 9y = 11 - 4x \]
02

Solve for y

Divide both sides of the equation by 9 to solve for y: \[ y = \frac{11 - 4x}{9} \]

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Key Concepts

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

Isolation of Variables
When solving a linear equation, the goal is to isolate the variable you are solving for. In this case, we want to isolate the variable **y**.

To do this, you need to move all other terms to the other side of the equation. This step is essential because it simplifies the equation, allowing you to solve for the desired variable.
Here鈥檚 how it's done in our example:

  • Start with the equation: \(4x + 9y = 11\).
  • Subtract 4x from both sides to isolate the term with y: \(9y = 11 - 4x\).
By isolating **y**, we make it easier to perform algebraic manipulations in the next steps.

Linear Equations
Linear equations are equations of the first degree, which means they form a straight line when graphed. These equations typically have variables raised to the power of one, and they appear in the general form \[ax + by = c\].

In our example, the equation \(4x + 9y = 11\) is a linear equation because both variables (x and y) are to the first power.

Some key features of linear equations:
  • They represent straight lines when graphed on a coordinate plane.
  • They have constant rates of change, meaning the slope (or rise over run) between any two points on the line is the same.
  • They appear in the form \(ax + by = c\) where a, b, and c are constants.
Understanding these characteristics can help you understand how to manipulate and solve them efficiently.

Algebraic Manipulation
Algebraic manipulation involves performing operations on equations to simplify or solve them. This includes adding, subtracting, multiplying, or dividing both sides of an equation by the same number.

Let鈥檚 finish solving our equation: \(9y = 11 - 4x\).
  • To solve for **y**, divide both sides of the equation by 9: \[y = \frac{11 - 4x}{9}\].
  • This isolates **y** on one side and expresses it in terms of **x**.
Remember these important rules during algebraic manipulation:
  • Whatever you do to one side of the equation, you must do to the other side.
  • Use inverse operations to isolate variables. For example, if a variable is being multiplied, use division to isolate it.
  • Keep equations balanced to maintain equality.
These principles ensure that your manipulations lead to correct solutions.

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Most popular questions from this chapter

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\}\) and \(D=\\{4\\} .\) Specify each set. See Examples I and 5. $$ B \cap \emptyset $$

Decide whether each statement is true or false. If it is false, explain why. The union of the set of rational numbers and the set of irrational numbers is the set of real numbers.

In baseball, winning percentage (Pct.) is commonly expressed as a decimal rounded to the nearest thousandth. To find the winning percentage of a team, divide the number of wins (W) by the total number of games played \((\mathrm{W}+\mathrm{L})\) Repeat Exercise 41 for the following standings for the Eastern Division of the National League. (a) Philadelphia (b) Atlanta (c) New York Mets (d) Washington $$ \begin{array}{|l|c|c|c|} \hline & {W} & {L} & {P c t} \\ \hline \text { Philadelphia } & {93} & {69} & {} \\ {\text { Florida }} & {87} & {75} & {.537} \\ {\text { Atlanta }} & {86} & {76} & {} \\ {\text { New York Mets }} & {70} & {92} \\ {\text { Washington }} & {59} & {103} \\ \hline \end{array} $$

Express each set in the simplest interval form. $$ [-1,2] \cup(0,5) $$

When a consumer loan is paid off ahead of schedule, the finance charge is less than if the loan were paid off over its scheduled life. By one method, called the rule of \(78,\) the amount of unearned interest (the finance charge that need not be paid) is given by $$ u=f \cdot \frac{k(k+1)}{n(n+1)} $$ In the formula, u is the amount of unearned interest (money saved) when a loan scheduled to run for n payments is paid off k payments ahead of schedule. The total scheduled finance charge is \(f .\) Use the formula for the rule of 78 to work Exercises \(37-40\) The finance charge on a loan taken out by Kha Le is \(\$ 380.50 .\) If 24 equal monthly installments were needed to repay the loan, and the loan is paid in full with 8 months remaining, find the amount of unearned interest.
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