Notes
Problems and Videos
For numbers 1-5, are the following linear?
\(\textbf{1)}\) \( 4x+2y=7 \)
\(\textbf{2)}\) \( 3x^2+5y=12 \)
\(\textbf{3)}\) \( \frac{2}{x}-y=7 \)
\(\textbf{4)}\) \( \frac{2x}{3}-y=7 \)
\(\textbf{5)}\) \( \sqrt{x}+\sqrt{y}=9 \)
\(\textbf{6)}\) Find the slope between the points \( (3,4) \) and \( (-7,2). \)
\(\textbf{7)}\) Find the \(x\)-intercept and \(y\)-intercept for \(y=3x+2. \)
\(\textbf{8)}\) Find the \(x\)-intercept and \(y\)-intercept for \(3x+4y=12.\)
\(\textbf{9)}\) Graph \( 2x-3y=6 \)
\(\textbf{10)}\) Graph \( y=-2x+3 \)
\(\textbf{11)}\) Graph \(y=.5x+1.5 \)
\(\textbf{12)}\) Find the equation of the line through the points \( (3,4) \) and \( (5,8). \)
\(\textbf{13)}\) What is the \(x\)-intercept of this line? \( y=3x-4 \)
See Related Pages\(\)
\(\bullet\text{ Graphing Linear Equations}\)
\(\,\,\,\,\,\,\,\,2x-3y=6 \) 
\(\bullet\text{ Slope Formula}\)
\(\,\,\,\,\,\,\,\,m=\frac{y_2-y_1}{x_2-x_1}\)
\(\bullet\text{ Net Change}\)
\(\,\,\,\,\,\,\,\,y_2-y_1\)
\(\bullet\text{ Point Slope Form}\)
\(\,\,\,\,\,\,\,\,y-y_1=m(x-x_1)\)
\(\bullet\text{ Parallel and Perpendicular Slope}\)
\(\,\,\,\,\,\,\,\,m_1=m+2,\,\,\,m_1=\frac{1}{m_2}\)
\(\bullet\text{ Distance Between a Point and a Line}\)
\(\,\,\,\,\,\,\,\,(3,4) \text{ and } y=\frac{3}{4}x−2\)
\(\bullet\text{ Finding x- and y- intercepts}\)
\(\,\,\,\,\,\,\,\,y=2x+4\)
In Summary
Linear equations can take the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope is a measure of the steepness of a line and is calculated by the rise over the run, or the change in y over the change in x. The y-intercept is the point where the line crosses the y-axis.
Linear equations and slope are typically covered in a high school algebra or geometry class.
A fun fact about linear equations and slope is that the slope of a line can be used to determine whether two lines are parallel or perpendicular. If the slopes of two lines are the same, the lines are parallel. If the slopes of two lines are negative reciprocals of each other, the lines are perpendicular.
Linear equations and slope have many real-world applications. For example, they can be used to model and analyze data in fields such as economics and engineering. They can also be used to solve practical problems such as calculating distances and determining the rate of change of a quantity over time.
The concept of linear equations and slope was first developed by the French mathematician René Descartes in the 17th century. He is credited with introducing the use of coordinates to represent points in space, which paved the way for the development of graphing techniques.