Solving simultaneous equations - AQASimultaneous examples with no common coefficients
Algebraic skills are required to find the values of letters within two or more equations. If two or more equations have the same variables and solutions, then they are simultaneous equations.
Solving simultaneous examples with no common coefficients
Some pairs of simultaneous equations may not have any common coefficientThe amount that a letter has been multiplied by. In the example of 3a, the coefficient of a is 3 because 3 x a = 3a..
For example, the simultaneous equations \(3a + 2b = 17\) and \(4a - b = 30\) have no common coefficient as the coefficients of \(a\) are 3 and 4, and the coefficients of \(b\) are 2 and -1.
Remember that a common coefficient is needed, regardless of sign. This means that -1 and 1 would be seen as a common coefficient.
In examples like this, one or both equations must be multiplied to create a common coefficient.
Create a common coefficient for either \(a\) or \(b\). In this case, making a common coefficient for \(b\) will be easier, as \(-b\) can be doubled to create \(-2b\), which will be a common coefficient throughout the equations.
Multiply the bottom equation to create a common coefficient of \(2b\).
Substitute the value of \(a\) into one of the original equations to find the value of \(b\).
\(3a + 2b = 17\) (when \(a = 7\))
Substitute \(a = 7\):
\(3 \times 7 + 2b = 17\)
\(21 + 2b = 17\)
Solve the equation by using inverse operationInverse operations are opposite calculations often used in solving equations. To remove +9 from a sum, perform the inverse operation which is -9.. The opposite of +21 is 鈭21. Subtract 21 from both sides of the equation:
\(2b = -4\)
The inverse of multiplying by 2 is dividing by 2. Divide both sides by 2:
\(b = -2\)
Check the answers:
\(4a - b = 30\) when \(a = 7\) and \(b = -2\)
\(4 \times 7 - - 2 = 30\)
According to sign rules, two minus signs make a plus when they are next to each other.
\(28 + 2 = 30\)
\(30 = 30\)
The equation balances, so the answers are correct. The final answers are \(a = 7\) and \(b = -2\).