Question

What are the values of \(m, x_{1},\) and \(y_{1}\) in the equation \(y-(-1)=3(x-2) ?\)

Short answer

For the equation provided, the values are \(m=3\), \(x_{1}=2\), and \(y_{1}=-1\).

Step-by-step solution

Identify the slope \(m\)

In the equation given, we don't need to do any calculations to find the slope \(m\). It is the number that multiplies the \(x\) part, namely \(3\). Thus, the value of \(m\) is \(3\).

Extract the \(x\) value

The \(x\) value, or \(x_1\), is the number that follows \(x\) within the parenthesis. It has a 'negative' sign in front of it in the equation. However, in this exercise, it's written as \(x-2\), which means \(x_1\) is actually \(2\).

Find the \(y\) value

Similar to \(x_1\), the \(y\) value, or \(y_1\), is the number after \(y\) in the parenthesis. However, the values within the equation may be given as a double negative. In this case, the given equation is written as \(y-(-1)\), implying the value for \(y_1\) is \(-1\).

Key concepts

Slope of a Line

Understanding the slope of a line is crucial when working with linear equations. The slope, denoted as 'm', measures the steepness, incline, or grade of a line. The mathematical definition of slope is the ratio of the vertical change between two points (rise) to the horizontal change between the same two points (run). In simpler terms, slope describes how much the 'y' value (vertical) changes for every one-unit increase in the 'x' value (horizontal).

In the given exercise, the slope is directly provided in the point-slope form of the equation. The formula for identifying the slope from a linear equation in point-slope form, which is \(y - y_1 = m(x - x_1)\), is simply the coefficient of the \(x\) term. Following this rule, we identify that \(m = 3\), indicating that for each single-unit increase in \(x\), \(y\) increases by three units.

Linear Equations

Linear equations form the foundation for graphing straight lines and are expressions that set two linear expressions equal to each other. They can be represented in various forms, including the standard form \(Ax + By = C\) and the slope-intercept form \(y = mx + b\), where 'm' represents the slope and 'b' the y-intercept. Another important form is the point-slope form, which is particularly useful when you know a point on the line \(x_1, y_1\) and the slope 'm'.

The point-slope formula is \(y - y_1 = m(x - x_1)\). In the exercise, this form quickly gives us valuable information about the line's slope and a specific point through which it passes. It shows the relationship of any point \(x, y\) on the line to the known point \(x_1, y_1\) and the slope 'm'. To solve for the values of \(m, x_1,\) and \(y_1\), one simply identifies these values as represented in the point-slope formula.

Graphing Linear Equations

Graphing linear equations is a visual representation of solutions to those equations on the Cartesian plane. To graph a linear equation like the point-slope form used in our exercise, you need a starting point and the slope. The point \(x_1, y_1\) defines where the line will cross a particular point in the grid, and the slope 'm' determines the angle at which the line will rise or fall across the plane.

With the equation from the exercise \(y-(-1)=3(x-2)\), we plot the known point \(2, -1\) and then use the slope to find a second point. Since the slope is 3, we rise 3 units up for every 1 unit we move to the right. By connecting these two points with a straight line, we have successfully graphed the linear equation. Remember, any two points are enough to draw a line since a straight line is defined as the shortest distance between two points.