Question

How do you solve this system of equations \[5x+10y=15\] and \[2x+8y=10\]?

Answer 1

Hint: The given system of equations does not have a direct relationship between the coefficients of x and y variables terms. For example, the coefficients are neither the same nor they have opposite signs, so directly adding or subtracting them will not be of any use. We can surely make the coefficient of x or y in the two equations the same. But it will involve extra calculations, as we have to multiply one/ both equations by a fraction/ Integers. So, for this system, we will follow the standard steps to solve. As we know these steps involve finding a relationship and substituting it in the other equation.

Complete step by step solution:
We are given the two equations \[5x+10y=15\] and \[2x+8y=10\]. We know the steps required to solve a system of equations in two variables. Let’s take the first equation, we get
\[\Rightarrow 5x+10y=15\]
Subtracting \[10y\] from both sides of equation, we get
\[\Rightarrow 5x=15-10y\]
Dividing both sides of the above equation by 5, we get
\[\Rightarrow x=3-2y\]
Substituting this in the equation \[2x+8y=10\], we get
\[\begin{align}
  & \Rightarrow 2\left( 3-2y \right)+8y=10 \\
 & \Rightarrow 6-4y+8y=10 \\
\end{align}\]
Simplifying the above equation, we get
\[\Rightarrow 4y=4\]
Dividing both sides of above equation by 4, we get
\[\Rightarrow y=1\]
Substituting this value in the relationship between variables to find the value of x, we get
\[\begin{align}
  & \Rightarrow x=3-2(1) \\
 & \Rightarrow x=1 \\
\end{align}\]

Hence, the solution values for the system of equations are \[x=y=1\].

Note: We can check if the solution is correct or not by substituting the values we got in the given equations.
Substituting \[x=y=1\] in the first equation, \[LHS=5(1)+10(1)=15=RHS\]. Substituting \[x=y=1\] in the second equation, \[LHS=2(1)+8(1)=10=RHS\]. Hence, as both equations are satisfied the solution is correct.