Question

Write and solve a system of linear equations: The larger of two numbers is 18 more than 5 timesthe smaller. If 3 times the larger number is increased by 4 times the smaller, the result is 16.Find the numbers by solving algebraically. Show your work including the equations

Answer 1

Let the larger number be

`=x`

Let the smaller number be

`=y`

Step 1: Represent the expression below as an equation

The larger of two numbers is 18 more than 5 times the smaller

`\begin{gathered} \text{Five times the smaller number } \\ =5* y \\ =5y \\ 18\text{ more than 5 times the smaller number will be} \\ x=5y+18\ldots\ldots\ldots\text{.(equation 1)} \end{gathered}`

Step 2: Represent the expression below as an equation

3 times the larger number is increased by 4 times the smaller, the result is 16.

`\begin{gathered} 3\text{ times the larger number will be} \\ =3* x \\ =3x \\ 4\text{ times the smaller number will be} \\ =4* y \\ =4y \\ \text{Therefore, the equation will be} \\ 3x+4y=16\ldots\ldots\ldots\text{.(equation 2)} \end{gathered}`

The positive sign was used in equation 2 because the question said increased

Step 3: Substitute equation (1) in equation (2)

`\begin{gathered} 3x+4y=16\ldots\ldots\ldots\text{.(equation 2)} \\ x=5y+18\ldots\text{.}(\text{equation 1)} \\ \text{hence,we will have} \\ 3(5y+18)+4y=16 \\ \text{expnading the brackets, we will have} \\ 15y+54+4y=16 \\ \text{collect similar terms,} \\ 15y+4y+54=16 \\ 19y+54=16 \end{gathered}`

Step 4: subtract 54 from both sides

`\begin{gathered} 19y+54=16 \\ 19y+54-54=16-54 \\ 19y=-38 \\ \text{divide both sides by 19} \\ (19y)/(19)=-(38)/(-19) \\ y=-2 \end{gathered}`

Step 5: Substitute the value of y=-2 in equation (1) to get the value of x

`\begin{gathered} x=5y+18 \\ x=5(-2)+18 \\ x=-10+18 \\ x=8 \end{gathered}`

Hence,

The larger number is = 8

The smaller number = -2