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2.4 B) Linear Simultaneous Equations: Elimination Method – Part 1
2.4 B) Linear Simultaneous Equations: Elimination Method – Part 1
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In the previous section, the simultaneous equations that we looked at had the same coefficients for y, and the signs of the coefficients were the same; the coefficients for the y’s in both of the equations were 2. In this section, we are going to have an example of a simultaneous equation whereby the coefficients are not the same. We answer questions like this by multiplying equations by a certain value to make the coefficients the same. This will make sense when we have a look an example.
Example 1
Solve the simultaneous equation:
In the previous section, the simultaneous equations that we looked at had the same coefficients for y, and the signs of the coefficients were the same; the coefficients for the y’s in both of the equations were 2. In this section, we are going to have an example of a simultaneous equation whereby the coefficients are not the same. We answer questions like this by multiplying equations by a certain value to make the coefficients the same. This will make sense when we have a look an example.
Example 1
Solve the simultaneous equation:
Neither of the coefficients of A or B are the same, regardless of sign. This means that we must create a common coefficient and we do this by multiplying an equation by a certain value. Before we do this, we must choose what coefficient we wish to make the same. I am going to make the coefficient of B the same (ignoring the signs). The coefficient of B in the first equation is 2, and the coefficient in the second equation is -4. To make the coefficients of B the same (regardless of sign), I can multiply the first equation by 2 (because 2 x 2 is 4, which will give me the same coefficient for B). We need to be really careful to ensure that we multiply every term in the first equation by 2. The first equation becomes:
The two equations are now:
We now have the same coefficients for B (4 in equation 1 and -4 in equation 2). As the coefficients are of opposite signs, we must plus the two equations to eliminate the unknown B (remember Same Take, Opposite Plus). This results in the equation becoming:
By adding the equations, we have eliminated B, and we have the equation that 7A is equal to 14. We want to find the value of A and not 7A. Therefore, we divide both sides of the equation by 7.
The value of A is 2. We can find the value of B by subbing in the value of A into either of the equations (it does not matter, which equation we use to find the value of B). Let’s sub our value for A into the first equation.
The first step in finding the value of B is to move the 2 from the left side of the equation to the right. We are able to do this by taking 2 from both sides of the equation.
We want to find the value of B and not 2B. Therefore, we need to divide both sides of the equation by 2.
We can check that we have the correct value for A and B by subbing them into the second equation.
This equation holds, thus meaning that we have the correct values for A and B; A is 2 and B is 4.
Coefficients of A the Same
In this question, we made the coefficients of B the same. However, we could have made the coefficients of A the same instead. This would have given us the same answers for A and B as we found earlier. Let’s now prove that this is the case. Here were the equations that we were given at the start of the question.
In this question, we made the coefficients of B the same. However, we could have made the coefficients of A the same instead. This would have given us the same answers for A and B as we found earlier. Let’s now prove that this is the case. Here were the equations that we were given at the start of the question.
In order to make the coefficients of A the same (the coefficient in the first equation is 1 and in the second equation is 5), we find the lowest common multiple (LCM) for 1 and 5. The LCM of 1 and 5 is 5. The coefficient in the second equation is already 5, so we do nothing to this equation. The coefficient of A in the first equation is 1, so we multiply the first equation by 5 to get the coefficient of A to be 5. When we are multiplying this equation by 5, we need to make sure that we multiply all of the terms in the equation by 5. This results in the first equation becoming:
We now have two equations with the same coefficient for A.
The coefficients are the same sign, which means that we take (Same Take, Opposite Plus). When we are taking equations, we need to be really carful with the signs. This is because the taking of a negative number results in you adding that number. For example, 50 – -6 becomes 50 + 6, which is 56. Also, 10B – -4B becomes 10B + 4B, which is 14B.
We now have an equation with one unknown; we have eliminated A. We want to find the value of B and not 14B. Therefore, we need to divide both sides of the equation by 14.
This tells us that B is 4.
We can find the value for A by subbing the value for B into either of the equations. I am going to sub it into the first non-changed equation (it is best to use the original equation and not an equation that you have changed because if you have made a mistake when multiplying the equation, you will be using the numbers on an equation that is wrong. Whereas the original equation will always be the right equation).
We can find the value for A by subbing the value for B into either of the equations. I am going to sub it into the first non-changed equation (it is best to use the original equation and not an equation that you have changed because if you have made a mistake when multiplying the equation, you will be using the numbers on an equation that is wrong. Whereas the original equation will always be the right equation).
We find the value of A by moving the 8 from the left side of the equation to the right. We are able to do this by doing the opposite and the opposite of adding 8 is taking 8. Therefore, we take 8 from both sides of the equation.
Therefore, A is 2 and B is 4, which is exactly the same as we got earlier when we eliminated B.
Example 2
Sometimes it will be the case that we will have to multiply both of the equations in order to get the coefficient of an unknown the same. For example, suppose that we had the simultaneous equations below:
Sometimes it will be the case that we will have to multiply both of the equations in order to get the coefficient of an unknown the same. For example, suppose that we had the simultaneous equations below:
For these simultaneous equations, we can get the coefficients of the x’s or the y’s the same. Let’s get the coefficients of the y’s the same. We get the coefficient of the y’s the same by making the coefficients the lowest common multiple between the current coefficients. The current coefficients are 2 and 3, and the lowest common multiple for these numbers is 6. Therefore, we want to have 6y in both of the equations. We are able to do this by multiplying the first equation by 3 and multiplying the second equation by 2.
We now complete the question in the normal way. As the signs are the same for the y’s, we would take the equations (Same Take, Opposite Plus).
By taking the equations, we have eliminated the y. We want to find the value of x and not 17x. Therefore, we divide both sides of the equation by 17.
The value for x is 1. We now sub x as 1 into either of the original equations. I am going to use the first original equation.
The first step in finding the value of y is to move the 7 from the left side of the equation to the right. We are able to move the 7 by taking 7 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 tells us that y is 3 and x is 1. We can check that these values are correct by subbing them into the first original equation.
This equation works, which means that we have found the correct values for x and y; x is 1 and y is 3.