The Effects of Multiplying Equations on Solutions in a Consistent Linear System

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.

In a linear system of equations, consistency refers to the property that the system has at least one solution

In a linear system of equations, consistency refers to the property that the system has at least one solution. If a linear system is consistent, it means that the system of equations has a common point of intersection, meaning the lines or planes represented by the equations intersect at a specific point.

Now, suppose we have a consistent linear system with multiple variables and equations. If we multiply every equation in the system by a constant, let’s say c, then the system’s solutions are affected by this multiplication.

To see this, let’s consider a system of linear equations with two variables, x and y, represented by the equations:

Equation 1: a1x + b1y = c1
Equation 2: a2x + b2y = c2

If this system is consistent, it means that there exists a point (x, y) that satisfies both equations simultaneously.

Now, let’s multiply every term in Equation 1 by the constant c:

c(a1x + b1y) = c(c1)
ca1x + cb1y = cc1

Similarly, we can multiply every term in Equation 2 by the constant c:

c(a2x + b2y) = c(c2)
ca2x + cb2y = cc2

Now we have a new system of equations:

Equation 3: ca1x + cb1y = cc1
Equation 4: ca2x + cb2y = cc2

We can observe that the coefficients of x and y in each equation have been multiplied by c, and the constants on the right-hand side have been multiplied by c as well.

Now, let’s consider a solution (x, y) to the original system of equations. This means that when we substitute x and y into each equation, both equations are satisfied. Therefore, the following equations hold true:

a1x + b1y = c1
a2x + b2y = c2

Now, if we substitute the solution (x, y) into the new system of equations, we have:

ca1x + cb1y = cc1
ca2x + cb2y = cc2

Multiplying the first equation by c:

c(a1x + b1y) = c(c1)
ca1x + cb1y = cc1

We can see that this equation is identical to Equation 3 in the new system. Similarly, substituting (x, y) into the second equation gives us the same equation as Equation 4 in the new system.

This shows that when we multiply each equation in a consistent linear system by a constant c, the solutions to the new system can be obtained by multiplying the solutions from the original system by c. In other words, if (x, y) is a solution to the original system, then (cx, cy) is a solution to the new system.

This concept can be extended to systems with more variables and equations. The principle remains the same: multiplying every equation by a constant will produce a new consistent system with solutions obtained by multiplying the original system’s solutions by that constant.

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