A linear equation in two variables, often written as \( y = mx + b \), describes a line in the \( xy \)-plane. The line consists of all the solution pairs \( (x_0, y_0) \) that make the equation true.
For example, in the equation, , the line would pass through the points and because these ordered pairs are solutions to the equation, as is every other point on the line.
The coefficient is the slope of the graph, and the constant is the -intercept.
The slope of a graph is the ratio between the rate of change of and the rate of change of , so for two points on a graph and , slope between them is:
What is the -intercept of the line described by ?
If , then , and so , which means the -intercept is .
What is in the point , if it falls on the line through the points and ?
The slope of the line is , so for all and on the graph. This means we can take one of the given points and substitute in its values to find .
Therefore, , and the equation of the line is . This means the point satisfies the equation . Thus, .
The intersection of the line with the line through the points , , is a point . Find the sum of and .
First, we need to find the equation of the line through and . Since , the slope of the line must be 11. Further, since , the -intercept is .
Thus, our two equations are:
To find the point of intersection , we solve the system of equations above by equating the two 's.
Therefore, , and placing into either of the equations for gives us as the point of intersection. The answer, then, is .