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2.4 A) Linear Simultaneous Equations: Elimination Method – Part 1
2.4 A) Linear Simultaneous Equations: Elimination Method – Part 1
When we have one equation where there is more than one unknown, there will be an infinite number of solutions. For example, there are an infinite number of solutions for x and y in the equation 3y + 2x = 8. Some of the solutions for the equation are:
From the above examples, you can see that there are an infinite number of solutions to the equation. However, to say for certain what the values are for x and y that solve the equation, we must be given another equation that has the same values for x and y as the original equation. We can use both equations to obtain the values for x and y. The two equations are known as simultaneous equations.
To solve simultaneous equations, we use one of 3 methods. The 3 methods are:
The elimination method and substitution method are algebraic methods, and the graphical method is where we plot the equations that we are given on a graph and the value(s) for x and y are where the lines intersect one another.
- x = 1 and y = 2: 3(2) + 2(1) = 6 + 2 = 8
- x = 2 and y = 4/3: 3(4/3) + 2(2) = 4 + 4 = 8
- x = 4 and y = 0: 3(0) + 2(4) = 0 + 8 = 8
- x = -½ and y = 3: 3(3) + 2(-½) = 9 + -1 = 9 – 1 = 8
From the above examples, you can see that there are an infinite number of solutions to the equation. However, to say for certain what the values are for x and y that solve the equation, we must be given another equation that has the same values for x and y as the original equation. We can use both equations to obtain the values for x and y. The two equations are known as simultaneous equations.
To solve simultaneous equations, we use one of 3 methods. The 3 methods are:
- The elimination method (using STOP)
- The substitution method (only needed for the higher paper)
- Graphical method
The elimination method and substitution method are algebraic methods, and the graphical method is where we plot the equations that we are given on a graph and the value(s) for x and y are where the lines intersect one another.
The Elimination Methods
The elimination method works by making the coefficient of one of the unknowns the same (we are only looking at the number of the coefficient and not the sign of the coefficient. For example, the coefficients would be the same if we had 3x in one equation and -3x in the other equation).
After the coefficients of one of the unknowns are the same (ignoring sign), the next step is to add or subtract the two equations.
The easy way to remember this is to use the mnemonic STOP, which stands for Same Take, Opposite Plus.
The elimination method works by making the coefficient of one of the unknowns the same (we are only looking at the number of the coefficient and not the sign of the coefficient. For example, the coefficients would be the same if we had 3x in one equation and -3x in the other equation).
After the coefficients of one of the unknowns are the same (ignoring sign), the next step is to add or subtract the two equations.
- If the coefficients have the same sign (positive and positive [2x in the first equation and 2x in the second equation], or negative and negative [-5y in the first equation and -5y in the second equation]) we would minus one of the equations from the other equation.
- If the coefficients have opposite signs (one is a plus and the other is a minus [4x in the first equation and -4x in the second equation]), we would plus the two equations.
The easy way to remember this is to use the mnemonic STOP, which stands for Same Take, Opposite Plus.
This will all make a bit more sense after we have gone through a few examples.
Example 1 – Coefficient of the Same Sign
Solve the simultaneous equation:
Solve the simultaneous equation:
The first step in answering simultaneous equations is to find which unknown (if any) have the same coefficient. From the two equations, we can see that the coefficients for y are the same and they are of the same sign (they are both plus 2). As the signs are the same, we take the two equations (remember Same Take, Opposite Plus). We therefore get:
We have now found the value of x, which is 3. We can find the value for y by subbing 3 in for x in either of the equations. It does not matter which equation we use because we will get the same value for y by using both equations (providing we have carried out everything correctly). I am going to use the first equation.
We find the value of y by getting all of the y’s on one side of the equation and all of the numbers on the other side. There are more y’s on the left than the right, so it makes sense to have all of the y’s on the left and all of the numbers on the right. This means that we need to move the 6 that is currently on the left side of the equation to the right side. We are able to do this by doing the opposite, which means that we take 6 from both sides of the equation.
We want to find the value of y and not 2y. Therefore, we need to divide both sides of the equation by 2.
This gives the value for y of 5. We can check that the answer that we have is correct by subbing the values that we have found for x and y into the second equation (which is the equation that we did not use to find the value of y). When we do this, we get:
The values for x and y work for this equation. Therefore, for this simultaneous equation, x is 3 and y is 5.