3

y = x + 1  |  y = 4 + x  

2
1
(x, y)
Guided Practice
{
 +  x = y

 +  x = y

When the graph of a system of two linear equations shows parallel lines that are not the same line, the system has no solution. When two lines are parallel, they have the same slope:

When two linear equations in a system show the same slope but are actually equations for the same line, then the system has an infinite number of solutions.

The equations y = 2x + 4 and y = 2(x + 2) are the same equation. So, the system with these two equations has an infinite number of solutions.

The two equations 2p + 2 = 6 and p + 4 = 6 both have right-hand sides equal to .

They also both have the same solution:
p = .

The system of equations 2x + 2 = y and x + 4 = y has one solution: (, ).

There are three possibilities for a system of linear equations:

solution(s)

solution(s)

solution(s)

Graph a different system on the coordinate plane on the left by entering a system of equations that has only one solution. Enter the solution using integers and decimals.

Solution: (, )

Graph a different system on the coordinate plane on the left by entering a system of equations that has no solutions.

Press the ✔ when you have entered a system with no solutions.

Graph a system on the coordinate plane on the left with just one solution: (5, 5).

Press the ✔ when you have entered a system with the solution.

Graph a system on the coordinate plane on the left with just one solution: (–3, –8).

Press the ✔ when you have entered a system with the solution.

Graph a system on the coordinate plane on the left with just one solution: (12, –2).

Press the ✔ when you have entered a system with the solution.

Graph a system on the coordinate plane on the left with no solutions.

Press the ✔ when you have entered a system with no solutions.

Graph a system on the coordinate plane on the left with just one solution: (–5, 14).

Press the ✔ when you have entered a system with the solution.

Graph a system on the coordinate plane on the left with just one solution: (4, 0).

Press the ✔ when you have entered a system with the solution.

Graph a system on the coordinate plane on the left with just one solution: (0, –53).

Press the ✔ when you have entered a system with the solution.

Determine without graphing whether the system has no solutions, one solution, or an infinite number of solutions. Write "no", "one", or "infinite".

This system has solution(s).

Determine without graphing whether the system has no solutions, one solution, or an infinite number of solutions. Write "no", "one", or "infinite".

This system has solution(s).

Determine without graphing whether the system has no solutions, one solution, or an infinite number of solutions. Write "no", "one", or "infinite".

This system has solution(s).

Determine without graphing whether the system has no solutions, one solution, or an infinite number of solutions. Write "no", "one", or "infinite".

This system has solution(s).

Determine without graphing whether the system has no solutions, one solution, or an infinite number of solutions. Write "no", "one", or "infinite".

This system has solution(s).

You can determine the solution(s) to a system of equations using algebra.

If we multiply both sides of the 2nd equation by –2.5, we get

Then add the two equations. Yes, we can do this! Try to figure out why we can.

Now that we know y, we can substitute its value into one of the equations to solve for x.

The method for solving a system shown in the previous example is called elimination. You can also use substitution to solve a system.

Substitute the expression x – 1 for y in the first equation and solve for x.

x – 1 = –x + 3

After you have solved for x, you can solve for y to determine the full solution to the system.

Complete the process shown in the worked example to determine the value of x.

x =

So, the solution to the system with the equations y = –5x + 2 and y = –2x – 1 is (, ).

Check this solution to make sure it is correct.

Solve the system of equations.

First, add the two equations. Write the result.

y =

Determine the value of y. Then use it to determine the value of x.

Solution: (, )

Solve the system of equations.

Multiply both sides of the first equation by –2. Write the result.

–2y = x +

Add the two equations. The result is –1y = 2. Then, divide both sides by –1. Write the result.

y =

Use the value of y to determine the value of x.

Solution: (, )

Solve the system of equations.

Multiply both sides of the first equation by –1. Write the result.

x + y =

Add the two equations. The result is 2.5x = 2.5. Then, divide both sides by 2.5. Write the result.

x =

Use the value of x to determine the value of y. Write the solution using decimals.

Solution: (, )

Determine the solution of the system of equations using any method.

Solution: (, )

Determine the solution of the system of equations using any method.

Solution: (, )

Determine the solution of the system of equations using any method.

Solution: (, )

Determine the solution of the system of equations using any method.

Solution: (, )

Determine the solution of the system of equations using any method.

Solution: (, )

A system of two linear equations whose graphs intersect at a single point has one solution. A system of two linear equations whose graphs are parallel but not intersecting has no solutions. And a system of two linear equations whose graphs are the same line has infinite solutions.

Use what you know to answer the questions.

Write the two linear equations whose graphs intersect at (4, 5).

y = x + 1   |   y = 4 + x   |   y = 2x – 3

and

Write the two linear equations whose graphs are the same line.

–0.5x + 3 = y   |   x – 6 = –2y   |   –0.5x + 1 = y

and

Write the two linear equations whose graphs are parallel but not intersecting.

y = 0.35x + 4  |  y = –0.35x  |  20y – 7x – 200 = 0

and

Write the two linear equations whose graphs intersect at (0, 0).

–5x = y  |  y = 1 + 5x  |  y = 5x

and

Write the two linear equations whose graphs intersect at (–1, 0).

–x – y – 1 = 0  |  x – y + 1 = 0  |  x – y – 1 = 0

and

A plane ascends from 10,000 ft to 20,000 ft in 5 min. At the same time, a second plane descends from 20,000 ft to 10,000 ft. When do the planes reach the same elevation?

Write equations to represent each plane's elevation in feet (y) over time in minutes (x). Then solve the system.

{
 +  x = y

 +  x = y

The elevation of the ascending plane can be represented by y = 2000x + 10,000. The elevation of the descending plane can be represented by y = –2000x + 20,000. Add the two equations. Write the result. (Use commas for numbers greater than 999.)

2y =

Then solve the system using whole numbers and decimals.

Solution: (, )

A number y is 10 less than 3 times another number x. The sum of x and y is 26.

Solve a system of equations to determine the values of x and y.

Solution: (, )

I bought 4 hot dogs and 3 burgers and paid $27. You bought 8 hot dogs and 4 burgers for $44.

Solve a system to determine the price of one hot dog (h) and one burger (b): (b, h).

Solution: (, )

Corinne wants to take dance classes. One school charges $15 plus $5 per class. Another charges $5 plus $7 per class.

Solve a system to determine the number of classes Corinne could take (c) and pay the same amount (p) at each school: (c, p).

Solution: (, )

You have 21 coins consisting of dimes and quarters. The number of dimes is 3 more than twice the number of quarters.

Solve a system to determine the number of dimes (d) and quarters (q): (q, d).

Solution: (, )

Two lines on a coordinate plane can either intersect at exactly one point, never intersect (are parallel), or intersect at every point (be the same line).

If two lines intersect at one point, then the same (x, y) pair makes the equations of the lines true. For example, the lines represented by the equations y = 2x – 5 and y = x + 1 intersect at
(6, 7), because:

7 = 2(6) – 5 and 7 = 6 + 1

Write the ordered pair of the point of intersection of the lines represented by these two equations: y = 4x + 4 and y = 9x – 1.

(, )