Question

In Problems 33-38, a point on a line and its slope are given. Find the point- slope form of the equation of the line. $$ P=(1,2) ; m=3 $$

Short answer

\( y - 2 = 3(x - 1) \)

Step-by-step solution

Identify the point-slope form equation

The point-slope form of a linear equation is given by \( y - y_1 = m(x - x_1) \). Here, \((x_1, y_1)\) is a point on the line, and \(m\) is the slope of the line.

Substitute the given point and slope into the equation

From the problem, the given point is \((1, 2)\) and the slope is \(m = 3\). Substitute \(x_1 = 1\), \(y_1 = 2\), and \(m = 3\) into the point-slope form equation: \( y - 2 = 3(x - 1) \)

Simplify the equation (optional)

If needed, the equation can be simplified to slope-intercept form (optional): \( y - 2 = 3x - 3 \) Add 2 to both sides: \( y = 3x - 1 \)

Key concepts

Linear Equations

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. These equations form straight lines when graphed on a Cartesian plane. The general form of a linear equation in two variables is: \[ ax + by = c \], where \( a \), \( b \), and \( c \) are constants. For the solutions of these equations, we often look for the values of \( x \) and \( y \) that make the equation true.

Linear equations are powerful tools that help us model and solve real-world problems in various fields like physics, economics, and everyday life scenarios. To fully grasp the subject, it's essential to understand the concepts of slope, intercepts, and different forms of linear equations.

Slope-Intercept Form

The slope-intercept form is one of the most popular ways to represent a linear equation. It is given by: \( y = mx + b \), where:

• \( m \) is the slope of the line, representing how steep the line is. It's the ratio of the vertical change 'rise' to the horizontal change 'run' between two points on the line.
• \( b \) is the y-intercept, which is the point where the line crosses the y-axis.

For example, let's convert the point-slope form equation from the given problem to slope-intercept form. Starting with \( y - 2 = 3(x - 1) \), we simplify as follows:

• Distribute the slope 3: \( y - 2 = 3x - 3 \).
• Add 2 to both sides to isolate y: \( y = 3x - 1 \).

Now, the equation \( y = 3x - 1 \) is in slope-intercept form, where the slope \( m = 3 \) and the y-intercept \( b = -1 \). This form makes it straightforward to graph the line since you can start at the y-intercept and use the slope to find other points.

Substitution Method

The Substitution Method is used to solve systems of equations, where two equations are solved together to find the values of the variables. The basic idea is to solve one of the equations for one variable and then substitute that expression into the other equation. Here's how it works:

• Solve one of the equations for one variable in terms of the other.
• Substitute this expression into the other equation.
• Solve the resulting equation for the remaining variable.
• Substitute back to find the value of the first variable.

For instance, let's say we have the following system of equations: \[ y = 3x - 1 \] and another equation like \[ y = 2x + 4 \].
Using the Substitution Method:

• From the first equation, we already have \( y \) in terms of \( x \), which is \( y = 3x - 1 \).
• Substitute \( 3x - 1 \) for \( y \) in the second equation: \( 3x - 1 = 2x + 4 \).
• Solve for \( x \):\( x = 5 \).
• Substitute \( x = 5 \) back into the first equation to find \( y \): \( y = 3(5) - 1 \), so \( y = 14 \).

Thus, the solution to the system is \( x = 5 \) and \( y = 14 \). This method is particularly useful when one of the equations is easily solvable for one variable.