Linear Equations – Slope-Intercept Form | Point-Slope Form | Horizontal & Vertical Lines

Linear Equations Learning Objectives Determine the equation of a line given the slope and y-intercept Determine the equation of a line given the slope and any point Determine the equation of a line given any two points on the line After this lesson, you will be able to determine the equation of a line given the slope and y-intercept, determine the equation of a line given the slope and any point, and determine the equation of a line given any two points on the line.

Slope-Intercept Form Standard-form – an equation of a line in the form Ax + By = C Value of A is greater than or equal to 0 Value of A and B are not both 0 Value of A, B, and C are real number constants Ex) 3x + 2y = 6 Subtract 3x from both sides of equation 2y = –3x + 6 Then, divide both sides of equation by 2 for One way to graph a line is to find two points on the line and connect them. But there are other ways to graph lines. The standard-form equation of a line is Ax + By = C {a x plus b y equals c}, where A is greater than or equal to 0, A and B are not both 0, and A, B, and C are real number constants. An example of an equation in this form is 3x + 2y = 6 {three x plus two y equals six}. A linear equation is easier to work with if the y-term is isolated. To solve this equation for y, first, subtract 3x from both sides of the equation. Then, divide both sides of the equation by 2.

Slope-Intercept Form Lines can be graphed by identifying the transformations on the linear parent function, y = x Ex) Reflected over the y-axis Vertically stretched Vertically translated up the y-axis by 3 (0,3) is the y-intercept Slope is –3/2 One way to graph a line is to isolate the y-term and identify the transformations on the linear parent function. The linear parent function, y = x, has been transformed in several ways to get the equation y = –3/2x + 3 {y equals negative three over two x plus three}. Multiplying x by the constant –3/2 reflects the line over the y-axis (because the constant is negative) and vertically stretches it (because the absolute value of the constant is greater than 1), whereas adding a constant of 3 vertically translates the line up the y-axis. From the graph, the y-intercept of the equation is located at the point (0,3), and the slope is –3/2. Notice that when y is isolated and the equation is simplified, the y-intercept and the slope can be found in the equation. This form of a linear equation is called the slope-intercept form because the slope can be found in the equation next to x, and the y-intercept is the added constant.

Slope-Intercept Form Equation can be written quickly if slope and y-intercept are known Slope and y-intercept recognized immediately in equation and line can be graphed quickly The slope-intercept form of an equation of a line is in the form y = mx + b where m is the slope of the line, and b is the y-intercept. Slope-intercept form of a line The slope-intercept form of a line is given. One advantage of the slope–intercept form of a line is that the equation can be written quickly if the slope and y-intercept are known. Another advantage is that the slope and y-intercept are recognized immediately in the equation so the line can be graphed quickly.

Point-Slope Form When a slope is given without a y-intercept, an equation can be formed from a point on the line Must apply transformations to the linear parent function until the equation represents the desired line Ex) If slope is 1/2, then the graph of the linear parent function should be vertically compressed If the point on the line is (4,5), then translate the point to the right by four units and up by five units Value of y for the given point can be moved to the other side of the equation The slope–intercept form of a line is useful when the slope and y-intercept are known or can be found from a graph. However, when the slope is given but the y-intercept is not known, it is still possible to find an equation for the line from a given point on the line. To do so, apply transformations to the linear parent function until the equation represents the desired line. For instance, if the slope of a line is 1/2, then the graph of the linear parent function should be vertically compressed. If the point on the line is (4,5), then translate the point on the linear parent function at the origin to the right by four units and up by five units. The y-value of the given point can be moved to the other side of the equation. This form of the equation is called the point–slope form because both the slope and one point on the line can be identified in this equation.

Point-Slope Form Advantages of point-slope form Equation can be formed quickly from the slope and one point of a line Can find y-intercept by solving the equation for y and converting the equation into slope-intercept form The point-slope form of the equation of a line is y – y1 = m(x – x1) where m is the slope of the line, and (x1,y1) is a point on the line. Point-slope form of a line The point-slope form of a line is given. An advantage of using the point–slope form is that an equation can be formed quickly from the slope and one point of a line. The y-intercept is not needed in this form, but it can be found by solving the equation for y and simplifying to convert the equation into slope-intercept form or by evaluating the function at x = 0.

Point-Slope Form Example Ex) A line passes through the following two points, (1,4) and (5,7). Find the equation of this line in point–slope form and slope–intercept form. Analyze Formulate Find slope and substitute into point–slope equation, solve for y Determine Justify Slope is located to the left of the x-value in each form Evaluate Depending on the information given, certain forms can be created faster A line passes through the following two points, (1,4) and (5,7). Find the equation of this line in point–slope form and slope–intercept form. First, analyze the problem. The problem says to find the equation of a line in both the point–slope and slope–intercept forms, given two points on the line. Next, formulate a plan or strategy to solve the problem. In either equation, the slope must be found utilizing the ratio of the change in y over the change in x. Once the slope is found, substitute it into the point–slope equation with either one of the given points. Then, solve the point–slope equation for y to write the slope–intercept form of the line. Next, determine the solution to the problem. Find the slope using the two points.   Write in point–slope form using the slope and the point (1,4). Isolate y and simplify to find the slope-intercept form. Now that the solution has been determined, justify it. The point–slope form of the equation is y – 4 = 3/4(x – 1) {y minus four equals three fourths times x minus one}, and the slope-intercept form of the equation is y = 3/4x + 3¼ {y equals three fourths x plus three and one fourth}. Both forms of the equation require the slope. The slope is located to the left of the x-value in each form. Because the point–slope form lends itself more easily to a point on the line that is not the y-intercept, this equation is formed first from the given information. Then, by solving for y and simplifying, the slope-intercept form of the line is created. Last, evaluate the effectiveness of the steps, and the reasonableness of the solution. An equation of a line can be formed in several ways. Depending on the information given, certain forms can be created faster. It is possible to switch from the point–slope form to the slope–intercept form when needed.

Horizontal & Vertical Lines Sometimes linear equations do not appear to contain both x- and y-components Ex) y = 4 Two points on the line are (–3,4) and (5,4) Slope is Standard form is 0x + y = 4 Slope–intercept form is y = 0x + 4 Possible point–slope equation is y – 4 = 0(x – 5) Sometimes linear equations do not appear to contain both x- and y-components. For instance, the equation of a line may appear like y = 4 or x = 4. Notice that two points on the line y = 4 are (–3,4) and (5,4). Use these two points to find the slope of this line. This relationship, which produces a slope of 0, is true for every line that is horizontal. Because the y-values do not increase or decrease, the slope of a horizontal line is always 0. However, the equation of the line can still be written in standard form, slope-intercept form, and point-slope form because it does truly contain an x- and a y-component. The standard form of this equation is 0x + y = 4 {zero x plus y equals four} because every x-value on this line has the same y-value. The slope-intercept form is y = 0x + 4 {y equals zero x plus four} because the slope is 0. Finally, one of the possible equations using the point-slope form is y – 4 = 0(x – 5) {y minus four equals zero times x minus five}, showing that all y-values are 4 regardless of the x-value.

Horizontal & Vertical Lines Ex) x = 4 Two points on the line are (4, –2) and (4,3) Slope is Impossible to write equation of a vertical line in slope–intercept or point–slope form Standard form is x + 0y = 4 The graph shows the line x = 4. Notice that two points on this line are (4, –2) and (4,3). Use these two points to find the slope of this line. Division by 0 is always undefined, so the slope of a vertical line is always undefined. It is impossible to write the equation of a vertical line in slope–intercept or point–slope form because these involve the slope, and the slope is undefined. However, the standard form of this vertical line can be written as x + 0y = 4 {x plus zero y equals four}. Notice that there can be an infinite number of y-values for the one x-value of 4. Vertical lines are not functions: there is more than one y-value associated with each x-value, and a function cannot have multiple outputs for a given input.

Linear Equations Learning Objectives Determine the equation of a line given the slope and y-intercept Determine the equation of a line given the slope and any point Determine the equation of a line given any two points on the line You should now be able to determine the equation of a line given the slope and y-intercept, determine the equation of a line given the slope and any point, and determine the equation of a line given any two points on the line.