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Chapter 2: Problem 107

Solve each equation for \(y\) $$ -3 x+5 y=-15 $$

Short Answer

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

Step by step solution

01

- Move the term with \(x\) to the other side

Start by isolating the term with \(y\). To do this, add \(3x\) to both sides of the equation: \(-3x + 5y + 3x = -15 + 3x\) This simplifies to: \(5y = 3x - 15\)
02

- Solve for \(y\)

Now, divide each term in the equation by 5 to solve for \(y\): \( y = \frac{3x - 15}{5} \) This can be simplified further by splitting the fraction: \( y = \frac{3x}{5} - \frac{15}{5} \) And then simplifying the constants: \( y = \frac{3}{5} x - 3 \)

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Key Concepts

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

Isolating Variable
Isolating a variable in an equation means to get the variable on its own on one side of the equation. This is usually done by performing operations like addition, subtraction, multiplication, or division on both sides of the equation. Imagine you have some weights on each side of a balance scale and you want to remove everything except one weight on one side. In the exercise, to isolate the variable \(y\), we first had to deal with the \(-3x\) term. By adding \(3x\) to both sides of the equation, we effectively moved the \(-3x\) term to the other side and isolated the term that contains \(y\):

  • Original: \(-3x + 5y = -15\)
  • Add \(3x\) to both sides: \(5y = 3x - 15\)

With the term containing \(y\) now isolated, we can proceed to solve for \(y\).
Linear Equations
A linear equation is an equation between two variables that gives a straight line when plotted on a graph. It typically takes the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. In our example, \(-3x + 5y = -15\) is a linear equation.

To solve a linear equation, you often need to isolate the variable you are solving for. This involves using techniques like:
  • Adding or subtracting terms from both sides
  • Multiplying or dividing both sides by a constant

Once a linear equation is solved, it can be graphed as a straight line. The solution we found for \(y\) was \(y = \frac{3}{5}x - 3\). This is a line where the slope is \(\frac{3}{5}\) and the y-intercept is \(-3\).
Simplifying Fractions
Simplifying fractions involves making a fraction as simple as possible. This means reducing the numerator (top part) and the denominator (bottom part) to their smallest values that still represent the same value. Let's look at how we handle fractions in the equation solution:

After isolating \(5y\), we had: \(5y = 3x - 15\). To solve for \(y\), we needed to get rid of the 5 by dividing each term by 5:

  • \(y = \frac{3x - 15}{5}\) becomes \(y = \frac{3x}{5} - \frac{15}{5}\)

Next, simplify each fraction:

  • \(\frac{3x}{5}\) remains as it is since it can't be simplified further
  • \(\frac{15}{5}\) simplifies to 3 because 15 divided by 5 is 3

So the final simplified form is:\
\( y = \frac{3}{5}x - 3 \). Breaking down and simplifying fractions step-by-step makes equations easier to understand and solve.

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Most popular questions from this chapter

Two of the most watched episodes in television were the final episodes of \(M^{*} A^{*} S^{*} H\) and Cheers. The total number of viewers for these two episodes was about 92 million, with 8 million more people watching the \(M^{*} A^{*} S^{*} H\) episode than the Cheers episode. How many people watched each episode? (Data from Nielsen Media Research.)

The Toyota Camry and the Honda Civic were top-selling passenger cars in the United States in 2016 . Civic sales were 22 thousand less than Camry sales, and 756 thousand of the two cars were sold. How many of each car were sold? (Data from www.businessinsider.com)

The Toyota Camry and the Honda Civic were the top-selling passenger cars in the United States in \(2016 .\) Civic sales were 22 thousand less than Camry sales, and 756 thousand of the two cars were sold. How many of each car were sold? (Data from wwy, businessinsider,com

A party-length submarine sandwich that is 59 in. long is cut into three pieces. The middle piece is 5 in. longer than the shortest piece, and the shortest piece is 9 in. shorter than the longest piece. How long is each piece?

Solve each formula for the specified variable. \(M=C(1+r)\) for \(r\)
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