Question

Write the slope-intercept form of the equation of the line that passes through the two points. Graph the line. Label the points where the line crosses the axes. \((1,2),(4,-1)\)

Short answer

The slope-intercept form of the line that passes through the points (1,2) and (4,-1) is \(y = -x + 3\). The line crosses the y-axis at (0,3) and the x-axis at (3,0).

Step-by-step solution

Find the slope

The slope of the line \(m\) that passes through the points \((x_1,y_1)\) and \((x_2,y_2)\) is given by the formula \(m = (y_2 - y_1) / (x_2 - x_1)\). Substituting the given values, we get \(m = (-1 - 2) / (4 - 1) = -3/3 = -1\).

Find the y-intercept

The y-intercept \(b\) is found using the formula \(b = y - mx\), substituting the values from one of the points and the slope, we get \(b = 2 - (-1)*1 = 2 + 1 = 3\).

Write the equation of the line

The equation of a line in slope-intercept form \(y = mx + b\). Substituting the values for the slope and the y-intercept, we get \(y = -x + 3\).

Graph the line

Plot the y-intercept (0, 3) as a point on the y-axis. Use the slope -1 (rise/run) to find another point and draw the line passing through these points. The line crosses the x-axis at point (3,0).

Key concepts

Graphing Linear Equations

Graphing linear equations involves plotting points and connecting them to form a straight line. A linear equation in two variables, such as x and y, can be represented visually on a coordinate plane. Here's how you can graph the equation from our exercise, using slope-intercept form:

First, identify the y-intercept from the equation, which is the point where the line crosses the y-axis. For the equation given in the exercise, \(y = -x + 3\), the y-intercept is \(3\), so you would plot the point \(0, 3\). Next, use the slope, which is \( -1 \), to find another point. Starting from the y-intercept, move down 1 unit and right 1 unit (since the slope is negative) and plot this second point. Finally, draw a straight line through these two points, and you've graphed your linear equation!

Finding the Slope

The slope of a line represents the steepness and direction of a line on a graph and is a crucial element in understanding linear equations. To find the slope when given two points, such as \((1,2)\) and \((4,-1)\), use the formula \((m = (y_2 - y_1) / (x_2 - x_1)\)). This is simply the change in y divided by the change in x, often referred to as 'rise over run'.

Substituting the given points into the formula results in \(m = (-1 - 2) / (4 - 1) = -3 / 3 = -1\). This tells us that for every step to the right along the x-axis, the line will fall one step on the y-axis, indicating it has a negative slope.

Understanding the Y-Intercept

The y-intercept is the point where a line crosses the y-axis and is where the value of x is zero. It is represented as \(b\) in the slope-intercept form \(y = mx + b\). Knowing this point helps to graph the line quickly and understand the starting value of the function.

In our example, we found the y-intercept by plugging in the slope and the coordinates of one point into the equation \(b = y - mx\). We calculated the y-intercept to be \(3\), given by \(b = 2 - (-1) * 1\). This means when x is zero, y is three, providing you with the starting point for the graph on the y-axis.

Algebraic Problem-Solving

Algebraic problem-solving is a step-by-step process used to find the unknowns in mathematical equations. In the context of linear equations, it involves finding the slope, the y-intercept, and writing the equation in slope-intercept form, \(y=mx+b\).

By systematically applying formulas and algebraic manipulation, as seen in our exercise, we derived the equation of a line given two points. This involves performing operations such as subtraction and division to find the slope, and then using multiplication and addition or subtraction to find the y-intercept. The culmination of this problem-solving process presents the graph of a linear function—clear and ready for interpretation.