How To Scale A Linear System Of Equations For Solving: Step-By-Step Guide

If each equation is consistent linear system is multiplied through by a constant c, then all solutions to the new system can be obtained by multiplying solutions from the original system by c.

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Let’s assume we have a consistent linear system of equations in the form:

a₁x + b₁y = c₁
a₂x + b₂y = c₂

Then, let’s multiply both sides of each equation by a constant c:

c(a₁x + b₁y) = c(c₁)
c(a₂x + b₂y) = c(c₂)

Distributing the constant c, we get:

ca₁x + cb₁y = cc₁
ca₂x + cb₂y = cc₂

Now, let’s assume that (x₀, y₀) is a solution to the original system. This means that plugging in x₀ and y₀ into each equation results in true statements:

a₁x₀ + b₁y₀ = c₁
a₂x₀ + b₂y₀ = c₂

We can see that if we multiply each side by c, we get:

ca₁x₀ + cb₁y₀ = cc₁
ca₂x₀ + cb₂y₀ = cc₂

Which means that (cx₀, cy₀) is a solution to the new system, since:

ca₁(cx₀) + cb₁(cy₀) = cc₁
ca₂(cx₀) + cb₂(cy₀) = cc₂

Thus, we’ve shown that for any solution to the original system (x₀, y₀), multiplying it by c to get (cx₀, cy₀) will result in a solution to the new system. We can also see that any solution to the new system (x, y) can be obtained by dividing each coordinate by c, since:

ca₁x + cb₁y = cc₁
a₁(x/c) + b₁(y/c) = c₁/c
a₁(x/c) + b₁(y/c) = (cc₁)/c

Thus, all solutions to the new system can be obtained by multiplying solutions from the original system by c.

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