Problem 11 - Algebra 1

Question

Write an equation of the line in point-slope form that passes through the given point and has the given slope. $$ (7,-7), m=-8 $$

Step-by-Step Solution

Answer

So, the equation of the line in point-slope form that passes through the point (7,-7) and has slope -8 is \(y = -8x + 49\).

Step 1 - Substitute the given point into the point-slope formula

The point-slope form of a line is \(y - y_1 = m(x - x_1)\). This formula requires a point and the slope of the line. Given the point (7, -7) and the slope m = -8, substitute the point (x1 = 7, y1 = -7) and the slope m = -8 into the equation. Thus, our formula becomes \(y - (-7) = -8(x - 7).

Step 2 - Simplify the equation

Simplifying the equation \(y - (-7) = -8(x - 7)\), we get \(y + 7 = -8x + 56\).

Step 3 - Isolate y

To get the equation in the normal form (y = mx + b), we need to isolate y by subtracting 7 from both sides of the equation. This gives us the equation \(y = -8x + 49\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent straight lines. The general form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
In other words, this equation gives us a straight line when plotted on a graph.
Linear equations are essential because they depict relationships where change happens at a constant rate. They appear frequently in various real-life situations, like comparing costs over time or predicting outcomes at different points.
To form a linear equation, you need two main pieces of information: a slope and a point on the line.

• The point-slope form is one way to express linear equations. It uses the equation \(y - y_1 = m(x - x_1)\), requiring one point \((x_1, y_1)\) and a slope \(m\).
• Converting between point-slope and slope-intercept form \((y = mx + b)\) helps visualize and graph the equation.

Understanding linear equations is the first step in graphing and analyzing lines efficiently.

Slope

Slope is a crucial concept when dealing with linear equations. It describes the steepness and direction of a line.
Throughout mathematics, it's denoted by \(m\) in the common linear equation \(y = mx + b\).
The slope can be found by calculating the ratio of the change in \(y\) to the change in \(x\) between two points on the line: \(\frac{y_2 - y_1}{x_2 - x_1}\).

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls from left to right, like in the example with \(m = -8\).
• A zero slope indicates a horizontal line, as there is no change in \(y\).
• Undefined slope occurs in vertical lines, where \(x\) doesn't change.

The slope not only indicates how steep a line is but also plays a role in predicting future values by understanding past trends.

Graphing Lines

Graphing lines helps us visualize linear equations, making comprehension easier.
The first step is to understand the point-slope form, where you have a point and a slope to guide you.
Given a point \((7, -7)\) and a slope \(m = -8\), you start graphing by plotting the point on a coordinate grid. From there, use the slope to find other points.

• The slope \(-8\) tells us that for every unit moved to the right on the \(x\)-axis, the line moves down 8 units on the \(y\)-axis.
• This descent pattern helps you plot additional points and draw a straight line through them.

Graphing lines through this method offers a clear visual picture of the equation, illustrating the linear relationship between \(x\) and \(y\). It allows you to see how the slope and intercept values affect the line's position and steepness on the grid.