solve by cross multiplication method 4x+3y=28,9x-5y=64
Answers
Definitions
Simultaneous Equations - Also known as a system of equations, simultaneous equations are a set of equations containing multiple variables.
For example, the equations x + 2y = 10 and 3x - y = -3 form simultaneous equations.
Solution - A value or set of values that make the simultaneous equations true.
For example, the solution to the equations 2x + 3y = 15 and x - y = -10 is x = -3 and y = 7.
Simultaneous equations are used when trying to find the intersection of two lines (two equations) or three planes (three equations). If any of the equations are equivalent, there will be an infinite number of solutions. If none of the equations are equivalent, then either one unique solution or no solution exists. (The process of solving simultaneous equations and finding whether a solution exists will be discussed in much greater detail in a few paragraphs).
Infinite Solutions
Two equations with two unknowns do not always have a unique solution.
You may remember the axiom: if you have 2 equations and 2 unknowns, you can find a solution. While this is technically true, it can be easily manipulated to trip up students since equivalent equations are actually the same line. Thus, there are an infinite number of solution points. Since this can be confusing in the abstract, consider the following example:
Equation 1: x + 2y = 5
Equation 2: -10 + 4y = -2x
Equation 1, Multiplied by 2: 2x + 4y = 10
Equation 1, Multiplied by 2: 2x + 4y - 10 = 10 - 10
Equation 1, Multiplied by 2: 2x + 4y - 10 = 0
Equation 1, Multiplied by 2: 2x - 2x + 4y - 10 = 0 - 2x
Equation 1, Multiplied by 2: 4y - 10 = - 2x
Equation 1, Multiplied by 2: - 10 + 4y = - 2x
Notice that Equation 1 = Equation 2
One Solution
Equation 1: x - y = -10
Equation 2: 2x + 3y = 15
Equation 1, Rearranged: y = x + 10
Equation 1 Substituted into 2: 2x + 3(x + 10) = 15
Equation 1 Substituted into 2: 2x + 3x + 30 = 15
Equation 1 Substituted into 2: 5x + 30 = 15
Equation 1 Substituted into 2: 5x + 30 - 30 = 15 - 30
Equation 1 Substituted into 2: 5x = -15
Equation 1 Substituted into 2: x = -3
Equation 1, Solve for Y: 2(-3) + 3(y) = 15
Equation 1, Solve for Y: -6 + 3y = 15
Equation 1, Solve for Y: 3y = 21
Equation 1, Solve for Y: y = 7
Solution: x = -3, y = 7
No Solution
Equation 1: 6x + 2y = 10
Equation 2: 12x + 4y = 21
Equation 1, Multiplied by 2: 12x + 4y = 20
The first equation contradicts the second equation. Thus, there will be no solution for these equations as no x and y will satisfy both equations. This happens because the two equations are parallel lines but have different x-intercepts and thus never intersect.