Problem 26 - Algebra 1

Question

Graph the points and draw a line through them. Write an equation in slope- intercept form of the line that passes through the points. $$ (0,-5),(3,4) $$

Step-by-Step Solution

Answer

The equation of the line passing through the points (0,-5), (3,4) is \( y = 3x -5 \).

Step 1: Plot the points

Plot the given points (0,-5) and (3,4) on a graph.

Step 2: Calculate the slope

Use the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Substituting in the given points, we get \( m = \frac{4 - (-5)}{3 - 0} = \frac{9}{3} = 3 \).

Step 3: Identify the y-intercept

The y-intercept in this case is the y-coordinate of the point where the line crosses the y-axis, in this case -5.

Step 4: Write the equation

Substitute the found slope and y-intercept into the slope-intercept form of the equation \( y = mx + b \). The final equation becomes \( y = 3x - 5 \).

Key Concepts

Plotting Points

Plotting points on a graph is a fundamental skill in understanding mathematical concepts, particularly in graphing linear equations. Each point on a coordinate plane is determined by an ordered pair \( (x, y) \), where \( x \) is the horizontal value and \( y \) is the vertical value. To plot the points from the exercise, such as \( (0, -5) \) and \( (3, 4) \) , you start at the origin (0,0) and count along the x (horizontal) axis first, and then along the y (vertical) axis for each respective point. The first point, \( (0, -5) \) , would be a point on the y-axis five units below the origin, whereas the second point, \( (3, 4) \) , would be found by moving three units to the right and four units up from the origin. When these points are marked, you can easily visualize how the rest of the graph will fall into place as a straight line connecting these points plays a key role in the solution.

Calculating Slope

The slope of a line illustrates how steeply it rises or falls as you move from one point to another. Calculating slope is essential for understanding linear equations and their graphical representations. The slope is found using the formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \), with \( m \) representing the slope and \( (x_1, y_1) \) and \( (x_2, y_2) \) being the coordinates of two points on the line. In the given exercise, substituting the given point values results in \( m = \frac{4 - (-5)}{3 - 0} = \frac{9}{3} = 3 \). This calculation reveals that the line goes up by 3 units for every one unit it goes to the right, showing a fairly steep incline.

Y-Intercept

The y-intercept of a line is the point at which the line crosses the y-axis on a graph. This is particularly important in the slope-intercept form of a linear equation because it represents the constant term 'b' in the equation \( y = mx + b \). The y-intercept gives you a starting point from which the line can be drawn when plotting. It's also the value of \( y \) when \( x = 0 \), making it straightforward to locate on a graph. In our example, the y-intercept is \( (0, -5) \) because this is where the line crosses the y-axis, suggesting that if no x-values are present, \( y \) will be -5.

Linear Equations

Linear equations form the backbone of algebra and are easily recognizable by their straight-line graphs. They describe a relationship between two variables, usually \( x \) and \( y \) , where the exponent of both is 1. The standard form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope tells us the direction and steepness of the line, while the y-intercept specifies where the line crosses the y-axis. Using the information from the exercise, the slope \( m = 3 \) and the y-intercept \( b = -5 \) are substituted into this form, yielding the linear equation \( y = 3x - 5 \). This equation allows us to predict \( y \) values for any given \( x \) and vice versa, thus describing a precise linear relationship.