Systems of two linear equations can be used to model two practical conditions that must be satisfied simultaneously.

A solution can also be thought of graphically as the point(s) of **intersection** of the graphs of the equations (the points in common to all graphs).

A system of linear equations can have no solutions, infinitely many solutions, or one solution. Consider the graphs of each of the following situations:

The number of solutions can be determined by looking at the equations without graphing.

- One solution: the lines have different slopes so they intersect at one point
- No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect
- Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line

A solution to a system of equations in a given context is said to be **viable** if the solution makes sense in the context and **non-viable** if it does not make sense within the context, even if it would otherwise be algebraically valid.

Graph each system of equations and state the solution.

a

\begin{cases} y = \dfrac{3}{4}x - 6 \\ 4y = 3x - 24 \end{cases}

Worked Solution

b

\begin{cases} 6x + 3y = 18 \\ 12x + 6y = 24 \end{cases}

Worked Solution

Rodica and Yuwei purchased used cars at the same time. Rodica buys a car with 9000 miles on it and drives an average of 200 miles each week. Yuwei buys a car with 6000 miles on it and drives an average of 500 miles each week. Consider the equation that represents the number of weeks when Rodica's and Yuwei's cars will have the same mileage:

200x+9000=500x+6000

a

Rewrite the equation as a system of equations.

Worked Solution

b

Construct a table of values to show when the cars will have the same mileage.

Worked Solution

c

Graph the system of equations. Choose an appropriate scale for the axes.

Worked Solution

d

Identify the solution for the system of equations and intrepret it in the context of the problem.

Worked Solution

Bixia is saving up her quarters and dimes in a jar. She has a total of \$24.50 in 125 coins.

a

Write a system of equations that models this situation.

Worked Solution

b

Graph the system of equations. Use appropriate axes, labels, and scales.

Worked Solution

c

Interpret the solution to the system of equations.

Worked Solution

Tyson is saving money in order to purchase a new smartphone for \$800 when the latest model is released. He currently has \$350 saved up and is able to put away \$100 each month.

a

Write a system of equations to represent the situation.

Worked Solution

b

Sketch the two lines representing these equations on the coordinate plane.

Worked Solution

c

If the new phone is to be released in 5 months' time, determine if Tyson will be able to afford it on release.

Worked Solution

Gordiano made two trips to a flower shop to purchase roses and sunflowers. On his first trip, he purchased 4 roses and 4 sunflowers and paid \$12. The following day, Gordiano went back to the flower shop and purchased 12 roses and 8 sunflowers for \$16. Without graphing, determine the number of solutions.

\begin{cases}4x + 4y=12\\ 12x+8y=16 \end{cases}

Worked Solution

Idea summary

The solution to a system of linear equations is the ordered pair of the point of intersection of the lines.

Systems of equations may have one solution, no solutions, or infinitely many solutions.

Graphing a system can help to determine the number of solutions a system will have:

- One solution: the lines have different slopes so they intersect at one point
- No solution: the lines have the same slope and different y-intercepts so they are parallel and will never intersect
- Infinitely many solutions: the lines have the same slope and y-intercept so they are the same line