Problem 89 - Algebra for College Students

Question

Find the equation of line l in each case and then write it in standard form with integral coefficients. Line \(l\) goes through \((-1,-2)\) and is perpendicular to \(y=-3 x+7\).

Step-by-Step Solution

Answer

The equation of the line in standard form is \(x - 3y = 5\).

Step 1 - Identify the slope of the given line

The slope of the given line, which is in the form of slope-intercept equation \(y = mx + b\), is \(m = -3\).

Step 2 - Determine the slope of the perpendicular line

The slope of a line perpendicular to another is the negative reciprocal of the other line's slope. Therefore, for a slope of \(-3\), the perpendicular slope \(m_p\) will be \(\frac{1}{3}\).

Step 3 - Use the point-slope form of the line equation

The point-slope form of a line equation is given by \(y - y_1 = m(x - x_1)\), where \(m\) is the slope and \( (x_1, y_1)\) is the point the line goes through. Plugging in the slope \(\frac{1}{3}\) and the point \((-1, -2)\): \[ y - (-2) = \frac{1}{3}(x - (-1)) \] \[ y + 2 = \frac{1}{3}(x + 1) \]

Step 4 - Simplify the equation

Distribute the slope on the right side and simplify: \[ y + 2 = \frac{1}{3}x + \frac{1}{3} \]

Step 5 - Convert to standard form

To convert this to the standard form \Ax + By = C\ with integral coefficients, multiply all terms by 3 to clear the fraction: \[ 3(y + 2) = x + 1 \] \[ 3y + 6 = x + 1 \] Now, rearrange the terms to get all variables on one side and constants on the other: \[ x - 3y = 5 \]

Key Concepts

Slope-Intercept Form

In algebra, the slope-intercept form of a linear equation is a handy way to write the equation of a line. The formula is given by: \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept, the point where the line crosses the y-axis. The slope \(m\) is a measure of how steep the line is. You can find the slope by determining the change in y divided by the change in x between two points on the line. For example, the line equation \(y = -3x + 7\) has a slope of \(m = -3\). If you have the slope and y-intercept, you can easily plot the line on a graph. To transform an equation into this form, isolate the y variable and arrange it as \(y = mx + b\).

Point-Slope Form

The point-slope form of a line equation is particularly useful when you know a point on the line and the slope. The formula is: \ (y - y_1 = m(x - x_1))\ , where \(m\) is the slope and \((x_1, y_1)\) is the known point. To illustrate, if we have a slope of \(\frac{1}{3}\) and a point \((-1, -2),\) the equation becomes: \[y - (-2) = \frac{1}{3}(x - (-1)) \] \[y + 2 = \frac{1}{3}(x + 1)\]. By distributing the slope, you get: \[ y + 2 = \frac{1}{3}x + \frac{1}{3} \] The form is great for creating linear equations based on specific conditions such as a particular intersection point.

Standard Form

The standard form of a linear equation is quite useful in algebra and geometric problems. This form is written as: \[ Ax + By = C\] where \(A, B, \text{ and } C\) are integers, and \(A\) should be positive. To convert different equations into this form, you typically rearrange and combine terms to move all variables to one side and the constant term to the other side. From our exercise’s simplified equation: \[ y + 2 = \frac{1}{3}x + \frac{1}{3} \] Upon clearing fractions by multiplying through by 3, we obtained: \[ 3y + 6 = x + 1 \] Rearranging terms leads to: \[ x - 3y = 5 \] which is in standard form. This is advantageous for comparing equations and solving systems of linear equations.