Problem 79 - Algebra for College Students

Question

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 \(4 x+2 y=5\).

Step-by-Step Solution

Answer

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

Step 1 - 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.

Step 2 - 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.

Step 3 - 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.

Step 4 - 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.

Key Concepts

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’s a straightforward way to quickly identify these key components of a line’s equation.

In our example, we first rewrote the given line's equation, 4x + 2y = 5, into slope-intercept form to find its slope. Let’s 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.