Problem 86 - 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 \((1,-3)\) and is parallel to \(y=-3 x-5\).

Step-by-Step Solution

Answer

The equation of line \(l\) in standard form is \(3x + y = 0\).

Step 1: Identify the slope of the given line

The given line is \(y = -3x - 5\). The slope of this line, denoted by \(m\), is \(-3\).

Step 2: Use the point-slope form of the equation of the line

Since line \(l\) is parallel to \(y = -3x - 5\), its slope is also \(-3\). The point-slope form of the equation is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (1, -3)\). Substituting the known values, we get \(y - (-3) = -3(x - 1)\).

Step 3: Simplify the equation

Simplify the equation from the point-slope form to the slope-intercept form. Starting with \(y + 3 = -3(x - 1)\), we expand to get \(y + 3 = -3x + 3\).

Step 4: Rearrange terms to standard form

To convert the equation to standard form, we rearrange the terms to move all terms to one side, resulting in \(3x + y = 0\). This is the equation of line \(l\) in standard form with integral coefficients.

Key Concepts

slope-intercept form

The slope-intercept form of a line is one of the most convenient ways to express a linear equation. It is written as y = mx + b.Here, 'm' represents the slope of the line, and 'b' is the y-intercept, which is the point where the line crosses the y-axis.

• The slope 'm' tells us how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope means the line falls.
• The y-intercept 'b' is the value of y at the point where the line crosses the y-axis (where x equals 0).

For example, in the given exercise, the equation of the line is y = -3x - 5. Here, the slope is -3, indicating a steep, downward trend, and the y-intercept is -5.

point-slope form

The point-slope form is especially useful when you know a point on the line and its slope. The formula is given by y - y_1 = m(x - x_1), where (x_1, y_1) is a point on the line, and 'm' is the slope.

• This form allows you to quickly create the equation of a line when you're provided with a point and the slope.
• It can be easily transformed into other forms like slope-intercept or standard form.

In our solution, we used the point-slope form on the point (1, -3) and the slope -3:y - (-3) = -3(x - 1).This can be expanded and simplified to convert into other forms later.

standard form

The standard form of a linear equation is another common way to express it. The general format is Ax + By = C, where A, B, and C are integers, and A should be a non-negative integer.

• This form is useful for quickly identifying the x-intercept and y-intercept of a line.
• It's also the form preferred for solving systems of linear equations using methods like elimination.

In our example, after simplifying and rearranging, we found the line equation to be:3x + y = 0.Here, A = 3, B = 1, and C = 0, all of which are integers.

parallel lines

Parallel lines are lines in the same plane that never intersect. A key property of parallel lines is that they have the same slope. If two lines have different slopes, they will eventually intersect at some point.

• Given two lines with equations in slope-intercept form y = m_1x + b_1 and y = m_2x + b_2, they are parallel if and only if m_1 = m_2.
• Knowing this, we can easily determine if two lines are parallel by comparing their slopes.

In the exercise, the original line was y = -3x - 5 with a slope of -3. When finding the equation for a line parallel to it through the point (1, -3), we ensured our new line also had a slope of -3:y + 3 = -3(x - 1). That our final standard form equation, 3x + y = 0, retains the same slope confirms the lines are indeed parallel.