Question Video: Using a System of Linear Equations to Find Two Quantities given Their Difference and Ratio Mathematics • 7th Grade

Video Transcript

An area of land is shared between two people in the ratio 13 to 10. The first person share is 81 square metres larger than the second person’s. What is the total area of land?

In this question, we have a proportional relationship between the first person share and the second person share. When we have a proportion relationship between two values 𝐴 and 𝐵, we can say that 𝐴 sub one to 𝐵 sub one equals 𝐴 sub two to 𝐵 sub two. And 𝐴 sub one over 𝐵 sub one equals 𝐴 sub two over 𝐵 sub two. The values of 𝐴 sub one and 𝐵 sub one refer to the quantities of 𝐴 and 𝐵 in the first situation. And 𝐴 sub two and 𝐵 sub two refer to the values of the quantity in a second situation. So let’s start by taking our ratio of 13 to 10 and writing it in a fractional form like we can see in the definition which would be 13 over 10.

For the second equivalent fraction, we need a way to note that the first person share is 81 square metres larger than the second person’s. So let’s use a value 𝑥 to represent the size of the second person share of land. When the first person share is 81 square metres larger, we could write this as 𝑥 plus 81. So now, we need to solve this equation to find 𝑥. We can begin by taking the cross product giving us 10 times 𝑥 plus 81 equals 13 times 𝑥.

We then multiply the 10 by each term in the parenthesis, starting with 10 times 𝑥, which is 10 𝑥, and then 10 times 81, which is 810. And we can write our 13 times 𝑥 as 13 𝑥. We can continue then by subtracting 10𝑥 from both sides of our equation, giving us 810 equal three 𝑥, since 13𝑥 take away 10𝑥 is three 𝑥. To find 𝑥, we divide both sides of our equation by three. So 𝑥 equals 270 square metres. So now, we know that the second person share, the value 𝑥, is 270 square metres. To find the first person share, we would work out 270 plus 81, which is 351 square metres. We now need to work out the total area of land. So we add 351 and 270 giving us 621 square metres.