Substitution Method
Solve the following system of equations using the substitution method: \(2x + 3y = 12\) and \(x = y + 4\)
Addition/Elimination Method
Solve the following system of equations using the Addition/Elimination method: \(3x - 2y = 4\) and \(2x + 3y = -1\)
Determining Perpendicular Lines
Find the equation of the line perpendicular to the line \(2x - 3y = 6\) and passing through the point \((1, 2)\).
Graphing Method
Solve the system of equations using the graphing method: \(y = 2x + 3\) and \(y = -x + 1\)
Determining if the Point is a Solution
Determine if the point (2,5) is a solution to the system of equations: \(2x + 3y = 16\) and \(x + 4y = 18\)
Finding the Constant of Variation
Consider the system of equations \(3x - y = 12\) and \(kx - 2y = 24\), where \(k\) is the constant of variation. If the system has no solution, what is the value of \(k\)?
Dependent, Independent, and Inconsistent Systems
Solve the following system of equations: \(2x + 3y = 6\) and \(4x + 6y = 12\)
Finding the Union (or)
Find the union of the solutions to the following system of equations: \n 1. \(2x + y = 10\) \n 2. \(x - y = 3\)
Finding the Equation with Real Coefficients
Find the equations of the lines that pass through the point (1,2) and are tangent to the circle with equation \(x^2 + y^2 = 25\).
Finding a Direct Variation Equation
Find the direct variation equation of a system of equations where \(y = 3x + 2\) and \(y = 5x - 1\).
Finding the Slope for Every Equation
Solve the following system of equations, and find the slope for each equation: \[\begin{matrix} 2x + 3y = 6 \ 5x - 4y = -2 \end{matrix}\]
Finding a Variable Using the Constant of Variation
Given the direct variation equation \(y = kx\), where \(y = 8\) when \(x = 4\), and a system of equations where \(2x + 3y = 10\) and \(5x - y = 15\), find the value of \(k\) and the solutions to the system of equations.