Linear Functions and Points

Linear equations, characterized by their unvarying rate of change, are mathematical constructs that graphically manifest as straight lines. They are typically written in the form y = mx + b. Points, denoted by their (x, y) coordinates, denote precise positions on these lines and adhere to the equation of the line itself.

Calculating Slope and y-Intercept

Find the slope and y-intercept of the line that passes through the points (1, 3) and (4, 11).

Solving Linear Equations

Find the point where the line given by the equation 3x - 2y = 6 intersects the y-axis?

Solving Radical Equations

Solve the radical equation \( \sqrt{x + 1} = 3x - 2 \) and verify if the solution is a point on the line \( y = 2x + 1 \)

Finding the Midpoint of a Line Segment

Find the midpoint of the line segment between the points A(4, 7) and B(10, 15).

Distance Formula

Find the distance between the point (3, 4) and the line 2x - 3y = 6 using the distance formula.

Finding a Point's Quadrant

Given the linear function \(f(x) = 3x - 2\), in which quadrant of the Cartesian plane is the point \((4, f(4))\)?

Finding Equations Using the Slope-Intercept Formula

Find the equation of the line that has a slope of 3 and passes through the point (4,2) using the slope-intercept form.

Finding Equations Using Slope-Intercept

Find the equation of the line that passes through the points (2,3) and (4,7) using the slope-intercept form.

Finding Equations Using Two Points

Find the equation of the line that passes through the points (3,7) and (6,13).

Finding a Perpendicular Line Containing a Given Point

Find the equation of the line perpendicular to the line \( y = 3x + 2 \) and passing through the point \( (4, -2) \).

Finding the Slope

Find the slope of the line that passes through the points (3, 5) and (-1, -7).

Finding a Parallel Line Containing a Given Point

Find the equation of the line parallel to the line \(3x - 4y = 12\) and passing through the point \((2, -3)\).

Finding the Equation Using Point-Slope Form

Find the equation of the line that passes through the points (2,5) and (4,9) using the point-slope form.