Question

Write the equation of each straight line passing through the given points and make a graph. $$(-3,5) \text { and }(1,3)$$

Short answer

The equation of the line is \(y = -\frac{1}{2}x + \frac{7}{2}\). To graph the line, plot the points (-3, 5) and (1, 3) and draw a straight line through them.

Step-by-step solution

Find the Slope (m)

To find the slope of the line that passes through two points ewline (-3, 5) and (1, 3), we use the slope formula: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ ewline Substitute the given points into the formula: $$m = \frac{3 - 5}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}.$$

Use the Point-Slope Form

The point-slope form of the equation is given by: ewline $$y - y_1 = m(x - x_1)$$ ewline Choose one of the points, say (-3, 5), and the slope \(-\frac{1}{2}\) to write the equation: ewline $$y - 5 = -\frac{1}{2}(x + 3).$$

Simplify to Slope-Intercept Form

Expand and simplify the equation to get the slope-intercept form, \(y = mx + b\): ewline $$y - 5 = -\frac{1}{2}x - \frac{3}{2} \rightarrow y = -\frac{1}{2}x + \frac{7}{2}.$$

Draw the Graph

To graph the equation, plot the two given points and draw a straight line through them. The slope \(-\frac{1}{2}\) tells us that for every two units we move horizontally to the right, the line will move down one unit. The point \( \frac{7}{2} \) is the y-intercept.

Key concepts

Slope of a Line

The slope of a line is a measure of its steepness and is typically represented by the letter 'm'. It is calculated by finding the difference in the 'y' values of two points on the line (known as rise) and dividing it by the difference in the 'x' values (referred to as run). The formula is ewlineewline \[ m = \frac{y_2 - y_1}{x_2 - x_1} \].ewline When the slope is positive, the line inclines upwards as it moves from left to right. Conversely, a negative slope indicates a downward incline. If the slope is zero, the line is horizontal, indicating a constant y-value regardless of x. Understanding the concept of slope is crucial since it not only determines the angle of the line but also helps us perceive the relationship between variables in various real-world applications.

Point-Slope Form

The point-slope form is a convenient way of writing the equation of a line when you know the slope and a single point on the line. It's expressed as ewlineewline \[ y - y_1 = m(x - x_1) \].ewline In this formula, \((x_1, y_1)\) is the point on the line, and 'm' represents the slope. This form is particularly useful when we want to derive the linear equation without needing to calculate the y-intercept directly. It illustrates how every other point \((x, y)\) on the line is relative to a known reference point and the slope. If the point-slope form is not your final destination, you can always manipulate it algebraically to get to other forms, such as the slope-intercept form, which will be discussed in the next section.

Slope-Intercept Form

The slope-intercept form is another popular way to express the equation of a straight line. It emphasizes the slope and the y-intercept of the line and is shown as ewlineewline \[ y = mx + b \].ewline Here, 'm' is the slope, and 'b' is the y-intercept, which is where the line crosses the y-axis. For example, if we have \[ y = -\frac{1}{2}x + \frac{7}{2} \], the slope 'm' is \(-\frac{1}{2}\) and the y-intercept 'b' is \(\frac{7}{2}\). This form is particularly user-friendly when it comes to graphing since you have direct access to the two most critical pieces of information you need: the starting point at 'b' when x is 0, and the slope to determine the rise over run. Students often find it easier to visualize and graph a line with this form.

Graphing Linear Equations

Graphing linear equations involves plotting points and drawing lines on the coordinate plane. You can use various forms of the equation of a line, but the slope-intercept form is frequently the most straightforward for this task. To graph the line, start by plotting the y-intercept on the y-axis. Then, from this point, use the slope to determine the line's direction and steepness. If the slope is a fraction \(\frac{1}{2}\), you would rise 1 unit and run 2 units from the y-intercept. Always remember to plot several points to ensure accuracy and draw the line through them. For the given problem, after plotting the points \((-3, 5)\) and \((1, 3)\) and using the slope learned earlier, you would draw a straight line through these points to represent the linear equation.