Chapter 5.4, Problem 14E

To determine

ToSolve:The system of linear equations

  $15x-5y=-20$

  $-3x+y=4$

Answer to Problem 14E

The given system of linear equations has infinitely many solutions.

Explanation of Solution

Given:

The system of linear equations

  $15x-5y=-20$

  $-3x+y=4$

Concept Used:

In solving a system of linear equation by substitution method. The value of one of the unknown from any of the equation is substituted into another equation and the value of second unknown is calculated.

The calculated value of second unknown is then substituted into any of the equation to find the value of the first unknown.

Calculation:

Given the system of linear equations

  $15x-5y=-20\text{ - - - - - }\left(1\right)$

  $-3x+y=4\text{ - - - - - }\left(2\right)$

Using the Substitution method

From equation $\left(1\right)$

  $\begin{array}{c}15x-5y=-20\\ 15x-5y+5y=-20+5y\text{ Add 5}y\text{ to both sides}\\ 15x=-20+5y\\ \frac{15x}{5}=-\frac{20}{5}+\frac{5y}{5}\text{ Divide by 5 to both sides}\end{array}$

  $3x=-4+y$

Substituting the value of $3x$ from equation $\left(1\right)$ into the equation $\left(2\right)$ , we have

  $\begin{array}{c}-3x+y=4\text{ From equation }(2)\\ -\left(-4+y\right)+y=4\text{ Substitue }-4+y\text{ for }3x\\ 4-y+y=4\\ 4=4\end{array}$

Since, the $4=4$ is always true.

Therefore, the given system of linear equations has infinitely many solutions.

Conclusion:

The given system of linear equations has infinitely many solutions.