Finding and Analyzing Linear Equations

What is the equation of the straight line passing through the coordinates A(2,3) and B(8,6)?

What is the intersection point of the equations 4x + 2y = -8 and 4x - 3y = 4?

Which method can be used to find the intersection point of the equations 4x + 2y = -8 and 4x - 3y = 4?

What is the slope of the line that passes through the points A(2,3) and B(8,6)?

If the equation of a linear line is expressed in the form y = mx + b, what does 'm' represent?

Finding the Equation of a Straight Line

Slope-intercept form: y = mx + c, where m is the slope and c is the y-intercept.
Slope: (change in y) / (change in x) = (6 - 3) / (8 - 2) = 3/6 = 1/2.
Using point-slope form: y - y1 = m(x - x1), where (x1, y1) is a point on the line.
- Choosing point A (2,3): y - 3 = (1/2)(x - 2)
- Simplifying to slope-intercept form: y = (1/2)x + 2.

Determining the Intersection Point of Two Linear Equations

Substitution method:
- Solve one equation for either x or y.
- Substitute the expression into the other equation.
- Solve for the remaining variable.
- Substitute the value back into one of the original equations to find the other variable.
Method used for the given equations:
- Solving the first equation for x: 4x = -8 - 2y => x = -2 - (1/2)y.
- Substituting this expression for x into the second equation: 4(-2 - (1/2)y) - 3y = 4.
- Simplifying and solving for y: -8 - 2y - 3y = 4 => -5y = 12 => y = -12/5.
- Substituting y = -12/5 back into the equation x = -2 - (1/2)y: x = -2 - (1/2)(-12/5) => x = -2 + 6/5 => x = -4/5.
Intersection point: (-4/5, -12/5).

This quiz focuses on deriving the equation of a straight line using slope-intercept and point-slope forms. It also covers methods to determine the intersection point of two linear equations through substitution. Test your understanding of these essential algebraic concepts.