Algebra Review: Systems of Linear Equations
Use With Lesson 4-6
You can solve a system of equations in two variables by using substitution.
Example 1
Algebra Solve the system.
y
=
3
x
+
5
y
=
x
+
1
table with 2 rows and 2 columns , row1 column 1 , bold italic y equals , column 2 3 bold italic x plus 5 , row2 column 1 , bold italic y equals , column 2 bold italic x plus 1 , end table
y
=
x
+
1
Start with one equation
.
3
x
+
5
=
x
+
1
Substitute
3
x
+
5
for
y
.
2
x
=
−
4
Solve for
x
.
x
=
−
2
table with 4 rows and 3 columns , row1 column 1 , y , column 2 equals x plus 1 , column 3 cap startwithoneequation . . , row2 column 1 , 3 x plus 5 , column 2 equals x plus 1 , column 3 cap substitute . 3 x plus 5 , for , y . , row3 column 1 , 2 x , column 2 equals negative 4 , column 3 cap solvefor . x . , row4 column 1 , x , column 2 equals negative 2 , column 3 , end table
Substitute
−
2
negative 2 for x in either equation and solve for y.
y
=
x
+
1
=
(
−
2
)
+
1
=
−
1
y equals , x plus , 1 equals , open negative 2 close plus , 1 equals , minus 1
Since
x
=
−
2
x equals , minus 2 and
y
=
1
,
y equals 1 comma the solution is
(
−
2
,
−
1
)
.
open negative 2 comma negative 1 close . This is the point of intersection of the two lines.
The graph of a linear system with infinitely many solutions is one line, and the graph of a linear system with no solution is two parallel lines.
Example 2
Algebra Solve the system.
x
+
y
=
3
4
x
+
4
y
=
8
table with 2 rows and 2 columns , row1 column 1 , x plus y , column 2 equals 3 , row2 column 1 , 4 x plus 4 y , column 2 equals 8 , end table
x
+
y
=
3
Start with one equation
.
x
=
3
−
y
Solve the equation for
x
.
4
(
3
−
y
)
+
4
y
=
8
Substitute
3
−
y
for
x
in the second equation
.
12
−
4
y
+
4
y
=
8
Solve for
y
.
12
=
8
False!
table with 5 rows and 3 columns , row1 column 1 , x plus y , column 2 equals 3 , column 3 cap startwithoneequation . . , row2 column 1 , x , column 2 equals 3 minus y , column 3 cap solvetheequationfor . x . , row3 column 1 , 4 . open , 3 minus y , close . plus 4 y , column 2 equals 8 , column 3 cap substitute . 3 minus y , for , x . inthesecondequation . . , row4 column 1 , 12 minus 4 y plus 4 y , column 2 equals 8 , column 3 cap solvefor . y . , row5 column 1 , 12 , column 2 equals 8 , column 3 cap falsefactorial , end table
Since
12
=
8
12 equals 8 is a false statement, the system has no solution.
Exercises
Solve each system of equations.
-
y
=
x
−
4
y
=
3
x
+
2
table with 2 rows and 2 columns , row1 column 1 , y , column 2 equals x minus 4 , row2 column 1 , y , column 2 equals 3 x plus 2 , end table
-
2
x
−
y
=
8
x
+
2
y
=
9
table with 2 rows and 2 columns , row1 column 1 , 2 x minus y , column 2 equals 8 , row2 column 1 , x plus 2 y , column 2 equals 9 , end table
-
3
x
+
y
=
4
−
6
x
−
2
y
=
12
table with 2 rows and 2 columns , row1 column 1 , 3 x plus y , column 2 equals 4 , row2 column 1 , negative 6 x minus 2 y , column 2 equals 12 , end table
-
2
x
−
3
=
y
+
3
2
x
+
y
=
−
3
table with 2 rows and 2 columns , row1 column 1 , 2 x minus 3 , column 2 equals y plus 3 , row2 column 1 , 2 x plus y , column 2 equals negative 3 , end table
-
y
=
x
+
1
x
=
y
−
1
table with 2 rows and 2 columns , row1 column 1 , y , column 2 equals x plus 1 , row2 column 1 , x , column 2 equals y minus 1 , end table
-
x
−
y
=
4
3
x
−
3
y
=
6
table with 2 rows and 2 columns , row1 column 1 , x minus y , column 2 equals 4 , row2 column 1 , 3 x minus 3 y , column 2 equals 6 , end table
-
y
=
−
x
+
2
2
y
=
4
−
2
x
table with 2 rows and 2 columns , row1 column 1 , y , column 2 equals negative x plus 2 , row2 column 1 , 2 y , column 2 equals 4 minus 2 x , end table
-
y
=
2
x
−
1
y
=
3
x
−
7
table with 2 rows and 2 columns , row1 column 1 , y , column 2 equals 2 x minus 1 , row2 column 1 , y , column 2 equals 3 x minus 7 , end table
