Problem 11 - Algebra 1: Concepts and Skills

Question

Write in point-slope form the equation of the line that passes through the given points. $$ (0,-10) \text { and }(12,4) $$

Step-by-Step Solution

Answer

The equation of the line that is parallel to the given line \(y=x+5\), and passes through the point \((-1, -1)\), is \(y = x\).

Step 1: Determine the Slope of Given Line

The given line is \(y=x+5\). In slope-intercept form, the coefficient of \(x\) is the slope, so the slope of the given line is \(1\).

Step 2: Identify the Given Point

The given point through which the new line should pass is \((-1, -1)\). We will label these as \(x1\) and \(y1\) respectively.

Step 3: Apply Point-slope Form

Next, we will plug the slope from Step 1 and the coordinates from Step 2 into the point-slope formula to find the equation of the line. Using \(y - y1 = m(x - x1)\), we have \(y - (-1) = 1*(x - (-1))\). Simplifying this yields \(y + 1 = x + 1\).

Step 4: Simplify the Equation

Simplify \(y + 1 = x + 1\) to get the final equation of the line, which is \(y = x\).

Key Concepts

Parallel Lines in Algebra

Understanding parallel lines is crucial when dealing with linear equations in algebra. Two lines are considered parallel if they are in the same plane and never intersect, no matter how far they are extended. This implies that parallel lines have the same slope.

When writing equations for parallel lines, the primary step is to determine the slope of the existing line. For example, if we have a line with an equation in slope-intercept form, like \( y = mx + b \), the slope is represented by the coefficient \( m \). If another line is parallel to this, it will have the same value of \( m \), but potentially a different \( b \), which is the y-intercept.

To find an equation for a line parallel to a given line and passing through a specific point, you use the point-slope form, \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point. Since the slopes are identical for parallel lines, this step guarantees that the new line will indeed run parallel to the given line.

Slope-Intercept Form

The slope-intercept form is pivotal for understanding and writing the equations of lines. It is expressed as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) indicates the y-intercept. The slope \( m \) tells us how steep the line is—how much the line rises (or falls) for each unit you move to the right (horizontally). The y-intercept \( b \) shows where the line crosses the y-axis.

In the case of the exercise, the equation of the given line is \( y = x + 5 \), which is already in slope-intercept form. Here, the slope (\( m \)) is 1, and the y-intercept (\( b \)) is 5. This form is particularly useful since you can easily graph the line and find parallel or perpendicular lines by looking at the slope.

Writing Equations of Lines

Writing equations of lines is a fundamental skill in algebra that allows you to model and solve real-world problems. There are various forms of line equations, but one specifically useful form is the point-slope form. It's given by the formula \( y - y_1 = m(x - x_1) \).

To write an equation in this form, you need two pieces of information: the slope of the line and a point through which the line passes. From the exercise, we know that the slope of our new parallel line is 1 (since it's parallel to \( y = x + 5 \)), and it passes through the point (-1, -1). Substituting these into the point-slope form gives us the equation of the new line. This form is particularly valuable when you have a point and a slope (from a parallel or perpendicular line) and need to quickly write down the equation of a line without rearranging it into a different form.