Final answer:
A linear function can be represented by the equation y = a + bx, where b is the slope and a is the y-intercept. The slope of a line describes its steepness and can be calculated by the change in y divided by the change in x. The y-intercept is the point where the line crosses the y-axis.
Step-by-step explanation:
A linear function can be represented by the equation y = a + bx, where b is the slope and a is the y-intercept.
The slope of a line describes its steepness and can be calculated by the change in y divided by the change in x. For example, if the slope is 3, it means that for every increase of 1 in the x-coordinate, the y-coordinate increases by 3.
The y-intercept is the point where the line crosses the y-axis and can be found by setting x = 0 in the equation and solving for y.
To graph a line given the slope-intercept form of the equation (y = mx + b), you can plot the y-intercept as a point on the y-axis and then use the slope to find additional points to plot on the line.
When a linear function is in standard form (Ax + By = C), you can find the x and y-intercepts by setting one of the variables to 0 and solving for the other variable. The x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis.
To rewrite a linear function in standard form to slope-intercept form, you can manipulate the equation algebraically to isolate y. For example, if the equation is 2x - 3y = 6, you can solve for y to get y = (2/3)x - 2.
To create an equation from a real-world situation, you need to identify the relationship between two variables and express it mathematically. For example, if you know that the cost of a shirt is $15 plus $5 for every hour of labor, the equation could be y = 5x + 15, where y is the cost and x is the number of hours of labor.
To find the equation of a line in slope-intercept form when given a slope and a point on the line, you can substitute the values into the equation y = mx + b and solve for b. For example, if the slope is 2 and the point (3, 5) is on the line, the equation could be y = 2x - 1.
To find the equation of a line in slope-intercept form when given two points, you can use the slope formula to find the slope and then substitute one of the points into the equation y = mx + b to solve for b. For example, if the points are (2, 4) and (5, 7), the equation could be y = x + 2.
To determine if a point is a solution to a linear inequality, you can substitute the coordinates of the point into the inequality and see if the statement is true. For example, if the inequality is 2x + 3y < 10 and the point (2, 3) is given, you would substitute x = 2 and y = 3 to get 2(2) + 3(3) < 10, which is true.
To graph a linear inequality on the coordinate plane, you can first graph the corresponding line using the slope-intercept form, and then shade the region that satisfies the inequality. For example, if the inequality is y > 2x + 1, you would graph the line y = 2x + 1 as a dashed line and shade the region above the line.