Transcript x –1 - Humble ISD

Objective:
Solve systems of equations by substitution.
Solve systems of equations by elimination.
Use substitution to solve the system of equations.
y= x–1
x+y=7
Step 1 Solve one equation for one variable. The
first equation is already solved for y: y = x – 1.
Step 2 Substitute the expression into the other
equation.
x+y=7
Substitute (x –1) for y in the other
x + (x – 1) = 7 equation.
2x – 1 = 7
2x = 8
x=4
Combine like terms.
Step 3 Substitute the x-value into one of the
original equations to solve for y.
y=x–1
y = (4) – 1
Substitute x = 4.
y=3
The solution is the ordered pair (4, 3).
Use substitution to solve the system of equations.
2y + x = 4
3x – 4y = 7
2y + x = 4
x = 4 - 2y
Since the x variable has a coefficient
of 1 in the first equation, solve the first
equation for x.
3(4 - 2y)– 4y = 7
Substitute x = 4 - 2y into the second
equation.
12 – 6y – 4y = 7
-10y + 12 = 7
-10y= -5
y= 1
2
2y + x = 4
1
2  +x = 4
2
Substitute the value to
solve for the other
variable.
1+x=4
X=3
The solution is
Second part of the solution
You can also solve systems of equations with the elimination method. With
elimination, you eliminate of one of the variables by adding or subtracting
equations. You may have to multiply one or both equations by a number to
create variable terms that can be eliminated. Variable terms may be
eliminated if their coefficients are additive inverses.
The elimination method is sometimes called the addition method or linear
combination.
Use elimination to solve the system of equations.
3x + 2y = 4
4x – 2y = –18
Step 1 Find the value of one variable. Select a
variable to eliminate. In this case the y variables
have coefficients that are additive inverses.
The y-terms have opposite coefficients.
3x + 2y = 4
+ 4x – 2y = –18 Add the equations to eliminate y.
7x
= –14
x = –2
First part of the solution
Step 2 Substitute the x-value into one of the
original equations to solve for y.
3(–2) + 2y = 4
2y = 10
y=5
Second part of the solution
The solution to the system is (–2, 5).
3x + y = 1
2y + 6x = –18
Because isolating y is straightforward, use substitution.
3x + y = 1
y = 1 –3x
2(1 – 3x) + 6x = –18
2 – 6x + 6x = –18
2 = –18
Regardless of the method used, sometimes, all variables will
disappear and a false statement will result. In this case, the system
represents parallel lines and the number of solutions is zero. The
solution is the empty set.
IF the result is a true statement such as 4=4, then the system
represents coincident lines and the number of solutions is infinte.
The solution would be “All real numbers on the line y = ……” Be
sure to state one of the lines!
Three Scenarios
Description
# of Solutions
Solution
Intersecting
One
Parallel
Zero
(#, #) an ordered
pair
Empty Set—No
solution
Coincident
Infinite
R on the line
(state the line)