Question

Write the point-slope form of the equation of the line that passes through the two points. $$ (2,6),(5,1) $$

Short answer

The point-slope form of the equation is \(y = -\frac{5}{3}x + \frac{16}{3}\)

Step-by-step solution

Find the slope

To find the slope between the points (2,6) and (5,1), use the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\), where \(x_1 = 2\), \(y_1 = 6\), \(x_2 = 5\) and \(y_2 = 1\). So \(m = \frac{1 - 6}{5 - 2} = -\frac{5}{3}\).

Substitute slope and a point into the point-slope form

Now substitute \(m = -\frac{5}{3}\) and the point (2,6) into the point-slope form equation. The equation is \(y - y_1 = m(x - x_1)\). Substitute to get \(y - 6 = -\frac{5}{3}(x - 2)\).

Re-arrange equation into point-slope form

The equation \(y - 6 = -\frac{5}{3}(x - 2)\) can be expanded and written in a simplified form. After expanding, it becomes \(y = -\frac{5}{3}x + \frac{16}{3}\)

Key concepts

Slope Calculation

When you're encountering a linear equation, the concept of a slope is central to understanding how the line behaves on a graph. The slope represents the rate of change between any two points on the line, essentially describing steepness and direction. To calculate the slope, denoted by the symbol \( m \), you use the slope formula which is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

In our example, the two points given are (2,6) and (5,1). Here, to calculate the slope, subtract the y-coordinate of the first point from the y-coordinate of the second point and divide it by the subtraction of the x-coordinate of the first point from the x-coordinate of the second point. This gives us \( m = \frac{1 - 6}{5 - 2} = -\frac{5}{3} \). The negative sign in the slope tells us that the line is descending from left to right on the graph, indicating a decrease per positive step along the x-axis. Learning to find the slope is essential as it is a key component in various forms of linear equations which will be discussed in the next sections.

Linear Equations

Linear equations form the foundation of Algebra and can be represented in numerous ways, including the standard form, slope-intercept form, and point-slope form. A linear equation is an algebraic expression that forms a straight line when graphed on a coordinate plane. These equations follow the general format \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept, the point where the line crosses the y-axis.

The point-slope form, specifically, is useful when you have the slope of a line and a point through which the line passes. It's expressed as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the known point. This form is particularly helpful for quickly writing down the equation of a line and is a go-to method when solving algebra problems involving lines. In the given exercise, you use the point-slope form and insert the known slope and one point to obtain the equation of the line.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In the context of linear equations, these expressions form one side of the equation, typically representing the y-value in terms of x. The structure of an algebraic expression reflects the relationship between variables and can show how changing one variable affects another.

In the solution to our exercise, after you insert the slope and the coordinates of a known point into the point-slope form, you arrive at an algebraic expression describing y in relation to x. This expression, once simplified, gives you a clear depiction of how y behaves as x changes, which in this case, is a linear relationship. The simplification process may involve distributing the slope across the parentheses and combining like terms to achieve the neatest form possible, making the relationship between x and y easy to interpret.