The variables 𝑎 and 𝑏 are related by the formula 𝑎 multiplied by 𝑏 plus five is equal to 𝑘 multiplied by 𝑏 minus 19𝑎. Make 𝑏 the subject.
In order to make a variable, in this case 𝑏, the subject of a formula, we need to rearrange the formula so that that variable is on its own on one side. As there are 𝑏s on both sides of the equation to begin with, we start by distributing the parentheses. On the left-hand side, we multiply 𝑎 by 𝑏 and then 𝑎 by five. This gives us 𝑎𝑏 plus five 𝑎. We repeat this process on the right-hand side. When distributing the parentheses here, we multiply 𝑘 by 𝑏 and 𝑘 by negative 19𝑎. This gives us 𝑘𝑏 minus 19𝑘𝑎.
Two of our four terms contain a 𝑏, 𝑎𝑏 and 𝑘𝑏. We need to get these two terms on one side of the equation and the other two terms on the other side. We can do this by adding 19𝑘𝑎 and subtracting 𝑎𝑏 from both sides of the equation. On the left-hand side, the 𝑎𝑏s cancel and we’re left with five 𝑎 plus 19𝑘𝑎. On the right-hand side, we have 𝑘𝑏 minus 𝑎𝑏. In order to make 𝑏 the subject, we next have to factor or factorise the right-hand side. As 𝑏 is common in both terms, we can factor this out of the parentheses.
𝑘𝑏 divided by 𝑏 is equal to 𝑘 and negative 𝑎𝑏 divided by 𝑏 is negative 𝑎. Therefore, we have 𝑘 minus 𝑎 inside the parentheses. On the left-hand side, we still have five 𝑎 plus 19𝑘𝑎. Our final step is to divide both sides by 𝑘 minus 𝑎. On the left-hand side, we’re left with five 𝑎 plus 19𝑘𝑎 divided by 𝑘 minus 𝑎. On the right-hand side, we’re left with 𝑏. Rearranging the formula to make 𝑏 the subject gives us five 𝑎 plus 19𝑘𝑎 divided by 𝑘 minus 𝑎.