Question

The larger of two numbers exceeds twice the smaller by one. Three times the smaller exceeds the larger by 6. If Y is the larger number, what would be the system of equations that represent this problem

Answer 1

Smaller number is \(a\) and larger number is \(b\)
"The larger of two numbers exceeds twice the smaller by one" gives you:\ "Three times the smaller exceeds the larger by 6" gives you:\[3(a)=b+6\]You now have a system of equations. This can be solved multiple ways, but we will take the substitution route since one of the variables is already isolated.\[\color{red}{b=2a+1},\text{ }3a=\color{red}{b}+6\]As a result:\[3a=(\color{red}{2a+1})+6\]\[3a=2a+7\]Now we need to isolate the variable. Move all relevant expressions to the left of the expression.\[3a\color{mediumorchid}{-2a}=\cancel{2a\color{mediumorchid}{-2a}}+7\]\[a=7\]Your smaller number is 7. Now we will calculate the larger number:\[\color{olive}{a=7},\text{ }b=2\color{olive}{a}+1\]\\So, your smaller number is 7 and the bigger is 15.