Question

Find an equation of the line that passes through the points, and sketch the line. $$ (-3,-4),(1,4) $$

Short answer

The equation of the line that passes through the points (-3,-4) and (1,4) is \(y = 2x + 2\). It is a linear line moving upwards with a positive slope of 2.

Step-by-step solution

Calculate the Slope

First, we need to calculate the slope of the line which is given by the formula \(\frac {y_2 - y_1} {x_2 - x_1}\) where \((x_1, y_1)\) and \((x_2, y_2)\) are the given points. Substituting our given points gives us \(\frac {4 - (-4)} {1 - (-3)} = 2\). So, the slope of the line, m is 2.

Determine the equation of the line

Next, we utilize the point-slope form to determine the equation of the line. The point-slope form is given by \(y - y_1 = m (x - x_1)\). Substituting any one of our given points, let's pick \((1,4)\), and the calculated slope into the equation gives \(y - 4 = 2 (x - 1)\). Simplifying this equation gives \(y = 2x + 2\).

Sketch the line

The sketching of the line on the Cartesian plane requires marking the points and drawing a straight line through them. The line passes through the points (-3,-4) and (1,4)

Key concepts

Slope Calculation

Understanding the concept of slope is crucial when dealing with linear equations. Slope represents the rate of change between two points on a line, or in simpler terms, how steep the line is. The formula for calculating the slope 'm' of a line that passes through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{ y_2 - y_1 }{ x_2 - x_1 } \). This formula is derived from the rise over run concept of a line on a Cartesian plane.

For example, to find the slope of a line that passes through the points \( (-3, -4) \) and \( (1, 4) \), we substitute these coordinates into our slope formula, yielding \( \frac{ 4 - (-4) }{ 1 - (-3) } = \frac{ 8 }{ 4 } \), which simplifies to \( m = 2 \). A positive slope, like we have in this scenario, indicates that the line rises as it moves from left to right.

It's important to take care when calculating the slope to avoid simple arithmetic mistakes and to ensure that the points are substituted into the formula in the correct order. This step is fundamental because the slope must be accurate before it can be used in the equation of the line.

Point-Slope Form

Once the slope is known, the next step is to write the equation of the line using the point-slope form. This form is expressed as \( y - y_1 = m(x - x_1) \), where 'm' is the slope and \( (x_1, y_1) \) is a point on the line. The point-slope form allows us to create an equation based on a known point and slope, which can then be manipulated into other forms like the slope-intercept form or standard form.

For our current example, using the slope \( m = 2 \) and the point \( (1, 4) \) our equation becomes \( y - 4 = 2(x - 1) \). This can be simplified further to \( y = 2x + 2 \), which is now in slope-intercept form, showing clearly the slope and the y-intercept of the line.

The point-slope form is especially efficient when dealing with problems requiring the equation of a line from a point and slope, and it is often the first step before arriving at more familiar forms. It is also a direct way to translate the geometric understanding of a line into its algebraic expression.

Plotting a Line on Cartesian Plane

When it comes to visualizing linear equations, plotting the line on a Cartesian plane provides an image of the abstract concepts used in algebra. This plane consists of two number lines that intersect at right angles: the horizontal x-axis and the vertical y-axis. The point where they intersect, \( (0, 0) \), is called the origin.

To plot a line, you'll need a minimum of two points through which the line passes. For instance, using our points \( (-3, -4) \) and \( (1, 4) \) from the example, we place a dot at each of these points on the Cartesian plane. Then, we draw a straight line through both points, extending it past them to show that it continues indefinitely in both directions, unless otherwise indicated.

Sketching the line helps in understanding the concept of the slope, the line's directional change, and the intercepts with the axes. It's also a practical way to verify if certain points are located on the line or to check the accuracy of the equation derived algebraically. Always remember to label your axes, points, and the line itself to make the plot clear and useful for study or reference.