Concept Byte: Systems With Rational Equations

For Use With Lesson 8-6

You can solve systems with rational equations using some of same methods you used with linear systems.

Activity 1

Follow each direction to solve the system { y = x 3 x − 1 y = 1 x + 1 . left brace . table with 2 rows and 1 column , row1 column 1 , y equals . fraction x , over 3 x minus 1 end fraction , row2 column 1 , y equals . fraction 1 , over x plus 1 end fraction , end table . .

1. Set the expressions for y equal to each other.
2. Solve for x.
3. Check your answer by substituting in the original system.

Activity 2

Follow each direction to solve the system { x − 2 = 6 y y + 1 = x . left brace . table with 2 rows and 1 column , row1 column 1 , x minus 2 equals , 6 over y , row2 column 1 , y plus 1 equals x , end table . .

4. Solve each equation for y.
5. Set the resulting expressions equal to each other.
6. Solve for x.
7. Check your answer by substituting in the original system.

Exercises

Solve each system.

8. { y x 2 − 4 x + 3 = − 2 x − 2 y = 3 left brace . table with 2 rows and 1 column , row1 column 1 , fraction y , over x squared , minus 4 x plus 3 end fraction . equals negative 2 , row2 column 1 , x minus 2 y equals 3 , end table
9. { y = 1 x y = 3 4 − x 2 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , 1 over x , row2 column 1 , y equals . fraction 3 , over 4 minus , x squared end fraction , end table
10. { y = x 2 − 2 x − 2 y = x 2 + x − 6 x + 3 left brace . table with 2 rows and 1 column , row1 column 1 , y equals , x squared , minus 2 x minus 2 , row2 column 1 , y equals . fraction x squared , plus x minus 6 , over x plus 3 end fraction , end table
11. { y = x + 2 x 2 + 3 x + 2 + 2 y − 3 = x left brace . table with 2 rows and 1 column , row1 column 1 , y equals . fraction x plus 2 , over x squared , plus 3 x plus 2 end fraction . plus 2 , row2 column 1 , y minus 3 equals x , end table
12. Reasoning It is possible for the graph of a system of rational equations to include a point of intersection that is an extraneous solution? Explain.

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