Video Transcript
Solve the following system of equations: five 𝑥 minus two 𝑦 equals eight and four 𝑥 plus three 𝑦 equals 11.
We have three options available to us when solving a pair of simultaneous equations. We can use either of the algebraic methods elimination or substitution, or we can attempt a graphical method. Let’s approach this problem using the substitution method. In order to use this method, we need to rearrange one of the equations to give an expression for one variable, either 𝑥 or 𝑦, in terms of the other.
Let’s rearrange equation one to give an expression for 𝑦 in terms of 𝑥. So, in other words, we want to make 𝑦 the subject of our first equation. First, we can isolate the 𝑦-term by subtracting five 𝑥 from each side, giving negative two 𝑦 equals negative five 𝑥 plus eight. Then we can divide both sides of the equation by negative two so that the coefficient of 𝑦 on the left-hand side is now one. And on the right-hand side, we have negative five over negative two 𝑥, which simplifies to five over two 𝑥 minus four. So, equation one has been rearranged to 𝑦 equals five over two 𝑥 minus four. And 𝑦 is now the subject of this equation.
We now take this expression for 𝑦 in terms of 𝑥 and substitute it into the other equation. So instead of four 𝑥 plus three 𝑦 equals 11. We now have four 𝑥 plus three multiplied by five over two 𝑥 minus four equals 11. And this gives an equation in 𝑥 only.
The first step in solving this equation is to distribute the parentheses. So, we have four 𝑥 plus 15 over two 𝑥 minus 12 is equal to 11. We can add 12 to each side, giving four 𝑥 plus 15 over two 𝑥 is equal to 23. And as we then want to combine the 𝑥-terms, we can change this integer four into the fraction eight over two, so it has the same denominator as our other 𝑥-term. So, we have eight over two 𝑥 plus 15 over two 𝑥 is equal to 23.
Combining the like terms on the left-hand side then, we have 23 over two 𝑥 is equal to 23. We can now cancel a factor of 23 from each side by dividing both sides by 23. So, we have 𝑥 over two is equal to one. To solve this equation for 𝑥, we multiply both sides by two and we find that 𝑥 is equal to two.
So, we found the value of one of the two variables. To find the value of the other, we need to take this value we’ve found for 𝑥 and substitute it back into our expression for 𝑦. 𝑦 is equal to five over two 𝑥 minus four. So, substituting 𝑥 equals two, we have 𝑦 is equal to five over two multiplied by two minus four. That’s five minus four, which is equal to one.
We’ve found the values of 𝑥 and 𝑦 then, but we should check our answer. We can do this by substituting the values of 𝑥 and 𝑦 into whichever equation we didn’t use to calculate the second value. We used a rearranged form of equation one, so we’ll substitute into equation two. On the left-hand side of equation two, we have four 𝑥 plus three 𝑦. So, using 𝑥 equals two and 𝑦 equals one, we have four multiplied by two plus three multiplied by one. That’s eight plus three, which is equal to 11. As this is the same as the value on the right-hand side of equation two, this confirms that our values of 𝑥 and 𝑦 are correct.
So, we solved this system of equations using the substitution method, and we found that 𝑥 is equal to two and 𝑦 is equal to one.
