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Solving Pairs of Equations Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Solving Pairs of Equations Practice Problems

PROBLEM 1

Using the Cartesian plane to plot their graphs, what can be said about the solutions to the simultaneous equations y = x + 3 and ( x − 1) 2 + ( y + 2) 2 = 9?

SOLUTION 1

The graphs of these equations are shown in Fig. 6-14. The equation y = x + 3 has a graph that is a straight line, ramping up toward the right with slope equal to 1, and intersecting the y axis at (0,3). The equation ( x − 1) 2 + ( y + 2) 2 = 9 has a graph that is a circle whose radius is 3 units, centered at the point (1, −2). It is apparent that this line and circle do not intersect. This means that there exist no solutions to this pair of simultaneous equations.

 

The Cartesian Plane Solving Pairs of Equations How Many Solutions?

Fig. 6-14 . Illustration for Problem 1.

PROBLEM 2

Using the Cartesian plane to plot their graphs, what can be said about the solutions to the simultaneous equations y = 1 and ( x − 1) 2 + ( y + 2) 2 = 9?

SOLUTION 2

The graphs of these equations are shown in Fig. 6-15. The equation y = 1 has a graph that is a horizontal straight line intersecting the y axis at (0,1). The equation ( x − 1) 2 + ( y + 2) 2 = 9 has a graph that is a circle whose radius is 3 units, centered at the point (1, −2). It appears from the graph that the equations have a single common solution denoted by the point (1, 1), indicating that x = 1 and y = 1.

 

The Cartesian Plane Solving Pairs of Equations How Many Solutions?

Fig. 6-15 . Illustration for Problem 2

Let’s use algebra to solve the equations and see if the graph tells us the true story. Substituting 1 for y in the equation of a circle (because one of the equations tells us that y = 1), we get a single equation in a single variable:

( x − 1) 2 + (1 + 2) 2 = 9

( x − 1) 2 + 3 2 = 9

( x − 1) 2 + 9 = 9

( x − 1) 2 = 0

x − 1 =0

x = 1

It checks out. There is only one solution to this pair of simultaneous equations, and that is x = 1 and y = 1, denoted by the point (1, 1).

Practice problems for these concepts can be found at: The Cartesian Plane Practice Test.

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