/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 56 Solve each equation for \(y\). ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 56

Solve each equation for \(y\). \(5 x-3 y=12\)

Short Answer

Expert verified
y = -4 + \frac{5}{3}x

Step by step solution

01

Isolate the term with y

First, start by moving the term containing y to one side of the equation. Subtract 5x from both sides of the equation:\[ 5x - 3y - 5x = 12 - 5x \]This simplifies to:\[ -3y = 12 - 5x \]
02

Eliminate the coefficient of y

Next, divide both sides of the equation by -3 to solve for y:\[ y = \frac{12 - 5x}{-3} \]
03

Simplify the equation

Finally, simplify the fraction by distributing the negative sign in the denominator:\[ y = -\frac{12}{3} + \frac{5x}{3} \]This simplifies to:\[ y = -4 + \frac{5}{3}x \]

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Key Concepts

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

Isolating Variables
In linear equations, isolating the variable is a key step. This means getting the variable alone on one side of the equation. To do this, you need to move other terms to the opposite side. Let's take the equation given: \[5x - 3y = 12\] To isolate \(y\), first move the term containing \(x\) by subtracting \(5x\) from both sides: \[5x - 3y - 5x = 12 - 5x\] This simplifies to: \(-3y = 12 - 5x\). Now, \(y\) is isolated with its coefficient. This sets the stage for the next steps.
Transposing Terms
Transposing terms means moving them from one side of the equation to the other by performing operations like addition, subtraction, multiplication, or division. This helps isolate the variable you want to solve for. In our example, after moving \(5x\) to the right side, we had: \(-3y = 12 - 5x\). To solve for \(y\), we need to divide each term by \(-3\). This transposition of dividing by \(-3\) ensures \(y\) stands alone: \[y = \frac{12 - 5x}{-3}\]. Transposing terms correctly is crucial because it maintains the equality of the equation.
Simplifying Expressions
Simplifying expressions helps make equations easier to understand and solve. Once we have \[y = \frac{12 - 5x}{-3}\], we simplify the fraction further. First, distribute the negative sign in the denominator: \[y = -\frac{12}{3} + \frac{5x}{3}\]. Simplification involves breaking down complex terms and combining like terms to get the simplest form. After distributing the negative sign, we get: \[y = -4 + \frac{5}{3}x\]. This is our simplified equation for \(y\).
Linear Equations
Linear equations are equations of the first degree, which means they involve only the first power of the variable. They typically have the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our example, the equation started as: \[5x - 3y = 12\]. Linear equations can have one, none, or infinitely many solutions depending on the values of the coefficients. In solving linear equations, you'll often isolate variables, transpose terms, and simplify expressions. By understanding these steps and practicing them, you'll have a strong foundation for tackling any linear equation.

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Most popular questions from this chapter

Three kinds of tickets are available for a Cowboy Mouth concert: "up close," "in the middle,", "and "far out." "Up close" tickets" cost 10 dollar more than "in the middle"" tickets, while "in the middle" tickets cost 10 dollar more than "far out" tickets. Twice the cost of an "up close" ticket is 20 dollar more than 3 times the cost of a "far out" ticket. Find the price of each kind of ticket.

Solve each system by the elimination method. $$ \begin{aligned} &0.5 x+3.4 y=13\\\ &1.5 x-2.6 y=-25 \end{aligned} $$

Solve each system by the elimination method. $$ \begin{aligned} &2 x-8 y=0\\\ &4 x+5 y=0 \end{aligned} $$

Solve equation. \(4 x-2(1-3 x)=6\)

Extend the method of this section to solve each system. Express the solution in the form \((x, y, z, w)\). $$ \begin{aligned} 3 x+y-z+2 w &=9 \\ x+y+2 z-w &=10 \\ x-y-z+3 w &=-2 \\ -x+y-z+w &=-6 \end{aligned} $$
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