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Chapter 3: Problem 79

Find the equation of line \(l\). Write the answer in standard form with integral coefficient with a positive coefficient for \(x .\) See Example 8. Line \(l\) goes through \((4,-2)\) and is parallel to \(4 x+2 y=5\).

Short Answer

Expert verified
The equation of the line in standard form is 2x + y = 6.

Step by step solution

01

- Determine the slope of the given line

The given line has the equation 4x + 2y = 5. We need to rewrite it in slope-intercept form (y = mx + b) to identify the slope. First, solve for y: 4x + 2y = 5 2y = -4x + 5 y = -2x + 5/2. The slope (m) of the given line is -2.
02

- Use the point-slope form of the equation

Since the line we need to find is parallel to the given line, it has the same slope of -2. The point-slope form of the line's equation is: y - y_1 = m(x - x_1). Given point is (4, -2) and the slope m is -2: y - (-2) = -2(x - 4). Simplify this.
03

- Simplify and convert to standard form

Simplify the equation from Step 2: y + 2 = -2(x - 4) y + 2 = -2x + 8 y = -2x + 6. Convert this to standard form by moving all terms to one side with x positive: 2x + y = 6.
04

- Validate the standard form

Ensure the equation is in standard form (Ax + By = C) with A > 0. The resulting equation is 2x + y = 6, where A = 2, B = 1, and C = 6, fulfilling this condition.

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Key Concepts

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

slope-intercept form
The slope-intercept form of an equation is one of the most used forms for linear equations. It is written as:

  • (y = mx + b)
Here, 'm' represents the slope of the line, and 'b' represents the y-intercept (the point where the line crosses the y-axis).
It鈥檚 a straightforward way to quickly identify these key components of a line鈥檚 equation.

In our example, we first rewrote the given line's equation, 4x + 2y = 5, into slope-intercept form to find its slope. Let鈥檚 go through this process step-by-step to solidify your understanding:

  • First, we subtracted 4x from both sides:
    2y = -4x + 5.
  • Then, we divided every term by 2 to solve for y:
    y = -2x + 5/2.
Now, it's evident that the slope (m) is -2.

Remember:
  • Slope (m) represents how steep the line is.
  • Y-intercept (b) represents the specific point where the line touches the y-axis.
point-slope form
The point-slope form of an equation is particularly helpful when you know a specific point on the line and its slope. The formula is:

  • (y - y_1 = m(x - x_1))
Here, (x_1, y_1) is a point on the line, and 'm' is the slope.

Using the point-slope form lets you plug in known values directly to write the equation of a line.

In our example, we were given the point (4, -2) and the slope -2. Plugging these into the point-slope formula, we get:

  • (y - (-2) = -2(x - 4))
Simplifying, we have:

  • (y + 2 = -2(x - 4))
Expanding the equation:

  • (y + 2 = -2x + 8)
To convert this to slope-intercept form, solve for y:

  • (y = -2x + 6)
standard form
The standard form of an equation of a line is written as:

  • (Ax + By = C)
      • Here, 'A', 'B', and 'C' are integers, and A should be a positive integer.

      Converting from slope-intercept or point-slope form to standard form involves rearranging terms and sometimes eliminating fractions.

      In our example, we already have the line equation in slope-intercept form: y = -2x + 6. To transform it into standard form entails:

      • First, add 2x to both sides to move all terms to one side:
        2x + y = 6.
      Now, it is successfully converted into standard form.

      Note:
      • Ensure 'A' is positive.
      • If needed, multiply the entire equation by -1 to flip the signs.
      In our case, 2 as 'A' is already positive, fulfilling the condition needed.

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Most popular questions from this chapter

Consider the graphs of \(y=2\) and \(x=3\) in the rectangular coordinate system. Explain why one of these relations is a function and the other is not.

Solve each problem. See Example 9. Carbon dioxide emission. Worldwide emission of carbon dioxide (CO \(_{2}\) ) increased linearly from 14 billion tons in 1970 to 26 billion tons in 2000 (World 91影视 Institute, www.wri.org). a) Express the emission as a linear function of the year in the form \(y=m x+b,\) where \(y\) is in billions of tons and \(x\) is the year. [ Hint: Write the equation of the line through \((1970,14) \text { and }(2000,26) .]\) b) Use the function from part (a) to predict the worldwide emission of \(\mathrm{CO}_{2}\) in 2010 .

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 \((4,0)\).

Find all intercepts for each line. Some of these lines have only one intercept. $$9 x+8 y=72$$

Let \(f(x)=3 x-2, g(x)=-x^{2}+3 x-2,\) and \(h(x)=|x+2|\). Evaluate each expression. See Example 9. $$ h(-3) $$
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