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Chapter 3: Problem 75

Find all intercepts for each line. Some of these lines have only one intercept. $$12+18 y=0$$

Short Answer

Expert verified
The y-intercept is (0, -2/3). There is no x-intercept.

Step by step solution

01

Rearrange the Equation

Start by rearranging the given equation to isolate the variable term. The given equation is \[ 12 + 18y = 0 \]Move the constant term (12) to the other side of the equation to isolate the term with the variable:\[ 18y = -12 \]
02

Solve for y

Now, solve for the variable y by dividing both sides of the equation by the coefficient of y (18):\[ y = \frac{-12}{18} \]Simplify the fraction:\[ y = -\frac{2}{3} \]
03

Determine Intercepts

The equation given is in the form of a vertical line equation when rearranged. Therefore, it only has a y-intercept. The y-intercept is the point where the line crosses the y-axis. Since this line does not depend on the x-variable, it does not have an x-intercept. The y-intercept is at\[ (0, -\frac{2}{3}) \].

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Key Concepts

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

Rearranging Equations
Rearranging equations is a fundamental skill in algebra. It involves shifting terms around to isolate one variable on one side of the equation. This practice makes it easier to solve for that variable.

For example, let's consider the equation given:
  1. Start with the original equation: 12 + 18y = 0

  2. To rearrange this, move the constant term to the other side by subtracting 12 from both sides:

  3. So, you get 18y = -12

In algebra, moving terms around correctly allows you to solve equations more easily. Remember, always perform the same operation on both sides to keep the equation balanced.
Solving for Variables
Once you rearrange the equation, the next step is to solve for the desired variable. This means isolating the variable completely on one side of the equation. In our example, we're solving for y.

Here鈥檚 how it's done:
  1. From the rearranged equation 18y = -12

  2. Divide both sides by the coefficient of y (18):
  3. This gives you: y = \( \frac{-12}{18} \)

  4. Simplify the fraction by dividing both numerator and denominator by their greatest common divisor (6):\( \frac{-12}{18} = -\frac{2}{3} \)

Now you've solved for y. It鈥檚 essential to perform each operation step-by-step and simplify your results.
Y-Intercept
Finding intercepts helps determine where the graph of an equation crosses the axes. The y-intercept is where the line crosses the y-axis, meaning the x-coordinate is zero at this point.

For our equation, since we only have a y variable, it simplifies directly to a y-intercept.

Here鈥檚 why:
  1. The final form y = -\( \frac{2}{3} \) informs us directly of where the line crosses the y-axis.

  2. Setting x to 0 results in the coordinate (0, -\( \frac{2}{3} \)), which is our y-intercept.

Lines that solely contain 'y' have no x-intercept, as they are parallel to the x-axis. Always look out for the variable types in the equation to find intercepts quickly.

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Most popular questions from this chapter

If Mildred earns \(\$ 14.50\) per hour, express her total pay \(P(h)\) as a function of the number of hours worked \(h .\) Find \(P(40)\).

Graph each pair of lines in the same coordinate system using the slope and y-intercept. $$ \begin{aligned} &y=\frac{2}{3} x+1\\\ &y=-\frac{3}{2} x+3 \end{aligned} $$

Explain the difference between a compound inequality using the word and and a compound inequality using the word or.

Exploration The intercept form for the equation of a line is 2 \(\frac{x}{a}+\frac{y}{b}=1\) where neither \(a\) nor \(b\) is zero. a) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{4}+\frac{y}{6}=1\) b) Find the \(x\) - and \(y\) -intercepts for \(\frac{x}{a}+\frac{y}{b}=1\) c) Write the equation of the line through \((0,3)\) and \((-5,0)\) in intercept form. d) Which lines cannot be written in intercept form?

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) has \(y\) -intercept \((0,5)\) and \(x\) -intercept \((4,0)\).
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