Question

If a linear system has four equations and seven variables, then it must have infinitely many solutions. True or False

Answer 1

Answer:

The statement is False.

Step-by-step explanation:

Consider the provided information.

If a linear system has four equations and seven variables, then it must have infinitely many solutions.

We need to determine the above statement is true or false.

The above statement is false, it could be inconsistent, and therefore have no solutions,

For example:

$x_1+x_2+x_3+x_4 +x_5+x_6+x_7=0\\x_1+x_2+x_3 =1\\x_4 +x_5 =1\\x_6+ x_7=1$

Hence, there is no solution.

Answer 2

If a linear system has four equations and 7 variables, then the system has infinitely many solutions, this is true.

When do systems have infinitely many solutions?

A system will have infinitely many solutions always that the number of variables is larger than the number of linearly independent equations.

For example, you can see cases where a system of 2 variables with 2 equations has infinitely many solutions.

That only happens when both equations represent the same line, thus in these cases, you have 2 variables and one equation.

Concluding, if you have 7 variables and 4 equations, you have more variables than equations, thus the system has infinitely many solutions.

If you want to learn more about systems of equations, you can read:

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