03Esimeqns-subst

toc = Simultaneous Equations =

Recall that to ** solve simultaneous equations ** means to find the coordinates of the point where their graphs intersect.

At this point, both equations are true at the same time.

Some simultaneous equations can be solved by plotting both graphs carefully on graph paper (See: Solving simultaneous equations using graphs).

OR

We can use technology (graphing program or CAS calculator) to solve simultaneous equations.

OR

We can use algebra.

We will learn two algebraic methods:
 * ** Substitution Method ** (use when one of the equations is in the form ** y = mx + c ** )
 * Elimination Method (use when both the equations are in the form ** ax + by = c ** )

Substitution Method
This method involve 4 steps:
 * 1) __**Substitute**__ one of the equations (the one in the form ** y = mx + c **) into the other equation.
 * 2) This gives an equation purely in terms of x, which we can solve for x.
 * 3) Substitute the value of x into one of the original equations and solve for y
 * 4) State the solution as a set of coordinates ** (x, y) **

** Example 1 **
Solve the following simultaneous equations using the substitution method:
 * y = 2x – 5
 * 2x + 4y = 10

__**Solution:**__

__**Checking Solution:**__

{We can check that our solution is correct by substituting ** (3, 1) ** into ** [2] **: the __other__ equation from the one we used to find y}

math \\ . \qquad 2x + 4y = 10 \qquad \{ \text{Substitute } \big(3, \; 1\big) \} \\ \\ . \qquad 2 \times 3 + 4 \times 1 = 10 \\ \\ . \qquad 6 + 4 = 10 \\ \\ . \qquad LHS = RHS math

SInce the Left Hand Side (LHS) __does__ equal the Right Hand Side (RHS), we know the solution is correct. **(3, 1)** ü

{If you get numbers that are __not__ equal, you know you have made a mistake somewhere}

** Example 2 **
Solve the following simultaneous equations using the substitution method
 * y = x + 1
 * y = 4x – 5

__**Solution:**__



__**Checking Solution:**__

{Substitute ** (2, 3) ** into equation ** [2] ** (the equation we didn't use to find y)} math \\ . \qquad y=4x-5 \qquad \text{Substitute } \big(2,\;3\big) \\ \\ . \qquad 3 = 4 \times 2 - 5 \\ \\ . \qquad 3 = 8 - 5 math

LHS = RHS so solution is correct
 * (2, 3) ** ü

Note: The most common error when solving simultaneous equations, is to stop after finding the x and forget to find the y-value as well. Remember that you are finding the coordinates of a point so you need both x and y values. State your solution as coordinates each time.

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