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Chapter 8: Problem 55

Solve each equation for \(y\). \(9 x-2 y=4\)

Short Answer

Expert verified
The solution is \( y = \frac{9x - 4}{2} \) or \( y = \frac{-4 + 9x}{2} \).

Step by step solution

01

- Isolate the term with y

Start by isolating the term with the variable y. To do this, subtract 9x from both sides of the equation: \[ 9x - 2y - 9x = 4 - 9x \] This simplifies to: \[ -2y = 4 - 9x \]
02

- Solve for y

To solve for y, divide both sides of the equation by -2: \[ y = \frac{4 - 9x}{-2} \] Simplify the fraction by distributing the negative sign: \[ y = \frac{-4 + 9x}{2} \] or \[ y = \frac{9x - 4}{2} \]

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Key Concepts

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

Isolating Variables
When solving linear equations, one of the core steps is isolating variables. This means getting the variable you are solving for on one side of the equation by itself. For instance, if we start with the equation 9x - 2y = 4, we need to move terms involving other variables to the opposite side. To isolate the term with y, subtract 9x from both sides: \[ 9x - 2y - 9x = 4 - 9x \] This simplifies the equation to: \[ -2y = 4 - 9x \] Here, we've managed to isolate the term involving y by removing the 9x term from the left-hand side. Isolating variables is an essential step in algebra because it simplifies the equation and makes it easier to solve for the unknown.
Solving for y
Once the y-term is isolated, the next step is solving for y. This generally involves getting y completely by itself on one side of the equation. Continuing from where we left off, we have: \[ -2y = 4 - 9x \] To isolate y, we need to divide both sides of the equation by -2: \[ y = \frac{4 - 9x}{-2} \] Next, distribute the negative sign to simplify the fraction: \[ y = \frac{-4 + 9x}{2} \] Or: \[ y = \frac{9x - 4}{2} \] So, y is now completely isolated and we have our solution. Solving for y like this is a common technique in algebra, making it essential for understanding more complex math concepts.
Linear Equations
A linear equation is an equation between two variables that produces a straight line when graphed. Its standard form is \(Ax + By = C\) where A, B, and C are constants. Linear equations have several important properties:
  • The solution is a set of points that form a straight line.
  • Each solution represents a unique point on the Cartesian plane.
  • They are fundamental in algebra and are used in various real-life applications.
In the equation \(9x - 2y = 4\), this is a linear equation because it can be rewritten in the form \(y = mx + b\), where m and b are constants. Solving it step-by-step like we did helps you find the specific relationship between x and y, making it easier to understand how changes in x affect y. This fundamental understanding of linear equations is crucial for topics like functions, graphing, and solving systems of equations.

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Most popular questions from this chapter

Solve system of equations by graphing. If the system is inconsistent or the equations are dependent, say so. \(2 x-y=4\) \(2 x+3 y=12\)

During the \(2008-2009 \mathrm{NHL}\) regular season, the Boston Bruins played 82 games. Their wins and overtime losses resulted in a total of 116 points. They had 9 more losses in regulation play than overtime losses. How many wins, losses, and overtime losses did they have that year? \(\begin{array}{|c|c|c|c|c|}\hline \text { Team } & {G P} & {W} & {L} & {O T L} & {P \text { oints }} \\ {\text { Boston }} & {82} & {} & {} & {} & {116} \\\ {\text { Montreal }} & {82} & {41} & {30} & {11} & {93} \\ {\text { Buffalo }} & {82} & {41} & {32} & {9} & {91} \\ {\text { Ottawa }} & {82} & {36} & {35} & {11} & {83} \\ {\text { Toronto }} & {82} & {34} & {35} & {13} & {81} \\\ \hline\end{array}\)

Solve each problem. How many gallons each of \(25 \%\) alcohol and \(35 \%\) alcohol should be mixed to get 20 gal of \(32 \%\) alcohol? $$\begin{array}{|c|c|c|}\hline \text { Gallons } & {\text { Percent }} & {\text { Gallons of }} \\ {\text { of Solution }} & {\text { (as a decimal) }} & {\text { Pure Alcohol }} \\ {x} & {25 \%=0.25} \\ {y} & {35 \%=0.35} \\\ {\text { 20 }} & {32 \%= 0.32}\end{array}$$

Pure acid is to be added to a \(10 \%\) acid solution to obtain \(54 \mathrm{L}\) of a \(20 \%\) acid solution. What amounts of each should be used?

Solve each system by the elimination method. $$ \begin{aligned} &2 x+3 y=0\\\ &4 x+12=9 y \end{aligned} $$
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