Question

How do you solve this system of equations \[0.05x+0.25y=66\] and \[0.15x+0.05y=72\]?

Answer 1

Hint: To solve the system of equations in two variables, we need to follow the steps given below in the same order, first we need to choose one of the equations to find the relationship between the two variables. This can be done by taking one of the variables to the other side of the equation. The next step is to substitute this relationship in the other equation to get an equation in one variable and solve this equation to find the solution value of the variable. Now, substitute this value in any of the equations to find the value of the other variable.

Complete step by step solution:
We are given the two equations \[0.05x+0.25y=66\] and \[0.15x+0.05y=72\]. We know the steps required to solve a system of equations in two variables. Let’s take the first equation, we get
\[\Rightarrow 0.05x+0.25y=66\]
Subtracting \[0.25y\] to both sides of the equation, we get
\[\Rightarrow 0.05x=66-0.25y\]
Multiplying this by 3 to both sides, we get
\[\Rightarrow 0.15x=3\left( 66-0.25y \right)\]
Substituting this in the equation \[0.15x+0.05y=72\], we get
\[\Rightarrow 3\left( 66-0.25y \right)+0.05y=72\]
Simplifying the above equation, we get
\[\Rightarrow y=180\]
Substituting this value in the relationship between variables to find the value of x, we get
\[\Rightarrow 0.05x=66-0.25(180)=21\]
Dividing both sides by \[0.05\] to the above equation, we get
\[\Rightarrow x=420\]

Hence, the solution values for the system of equations are \[x=420\And y=180\].

Note: Here, we multiplied the equation \[0.05x=66-0.25y\] by 3, because the coefficient of the x in the first equation and coefficient of x in the second equation are in the ratio \[1:3\]. Thus, by doing this it becomes easier to do substitution, and avoid extra calculations.