Chapter 9.7, Problem 26HP

To determine

To describe : a system of equations in two variables that has infinitely many solutions, both graphically and algebraically.

Answer to Problem 26HP

The system of equationsin two variables that has infinitely many solutions:

  $\begin{array}{l}x+5y=10\\ 2x+10y=20\end{array}$

The graph for the system of equations:

Explanation of Solution

Given information :

The system of equations to be described should haveinfinitely many solutions.

Calculation :

As per problem,

Consider the system of equations:

  $\begin{array}{l}x+5y=10\\ 2x+10y=20\end{array}$

Solve the firstequation for $x$ since its coefficient is $1$ .

  $\begin{array}{l}\text{}\text{}\text{}\text{}\text{}\text{}x+5y=10\text{ [Write the first equation]}\\ x+5y-5y=10-5y\text{ [Subtract 5}y\text{ from each side]}\\ x=10-5y\text{ [Simplify}]\end{array}$

Substitute $10-5y$ for $x$ in the second equation.

  $\begin{array}{l}2x+10y=20\text{ [Write the second equation]}\\ 2\text{(}10-5y\text{)}+10y=20\text{ [Substitute }x=10-5y\text{]}\\ 20-10y+10y=20\text{ [Distributive Property]}\\ \text{ 20}=20\text{ [Simplify]}\end{array}$

  $20=20$ is a true statement. The system of equations has infinite number of solutions.

Graph:

Graph the equations on the same coordinate plane.

Both equations have the same graph. Any ordered pair on the graph will satisfy both the equations. Therefore, there are infinitely many solutions for this system of equations.