I was just wondering if anybody could explain to me the steps in 'solving systems in substitution' and 'solving systems in substitution with a variable'. I'm majorly confuzled!!!!
It might help to post a specific problem so someone can show you the steps.
-5x+y=-3. 3x-8y=24
-5x+y=-3
+5x +5x
___________
Y=5x-3
3x-8(5x-3)=24
3x-40x+24=24
-37x+24=24
-24-24
___________
-37x=???
Of course, I'd be happy to explain the steps in solving systems of equations using the substitution method and the substitution method with a variable. Let's start with the basic substitution method.
The substitution method is a technique used to solve a system of equations by isolating one variable in one equation and substituting it into the other equation. Here are the steps involved:
1. Begin by solving one of the equations for one variable in terms of the other variable. Choose the equation that looks easiest to work with.
2. Substitute the expression obtained from step 1 into the other equation for the same variable. This creates a new equation with only one variable.
3. Solve the new equation from step 2 for the remaining variable.
4. Substitute this value back into either of the original equations to find the value of the other variable.
5. Once you have found the values of both variables, write the solution as an ordered pair (x, y).
To illustrate, let's look at an example:
Example: Solve the system of equations using substitution method:
Equation 1: 2x + y = 7
Equation 2: x - y = 3
Step 1: Solve Equation 2 for x in terms of y:
x = y + 3
Step 2: Substitute x = y + 3 into Equation 1:
2(y + 3) + y = 7
Step 3: Simplify and solve for y:
2y + 6 + y = 7
3y + 6 = 7
3y = 1
y = 1/3
Step 4: Substitute y = 1/3 into Equation 2:
x - (1/3) = 3
x = 3 + 1/3
x = 10/3
Step 5: The solution is (10/3, 1/3).
Now, let's move on to solving systems in substitution with a variable.
In some cases, the substitution method can be simplified by introducing a new variable. This method works well when one variable can be easily isolated in one equation, but not in the other. Here are the steps involved:
1. Solve one equation for one variable in terms of the other variable. Choose the equation that allows you to easily isolate a variable.
2. Substitute the expression obtained from step 1 into the other equation, creating a new equation with only one variable.
3. Solve the new equation for this variable.
4. Substitute the value obtained from step 3 into either of the original equations to solve for the second variable.
5. Write the solution as an ordered pair (x, y).
Let's look at an example to help clarify:
Example: Solve the system of equations using substitution method with a variable:
Equation 1: 3x + 2y = 10
Equation 2: 2x - y = -1
Step 1: Solve Equation 1 for x in terms of y:
x = (10 - 2y)/3
Step 2: Substitute x = (10 - 2y)/3 into Equation 2:
2((10 - 2y)/3) - y = -1
Step 3: Simplify and solve for y:
(20 - 4y)/3 - y = -1
(20 - 4y - 3y)/3 = -1
(20 - 7y)/3 = -1
20 - 7y = -3
-7y = -23
y = 23/7
Step 4: Substitute y = 23/7 into Equation 1:
3x + 2(23/7) = 10
3x + (46/7) = 10
3x = 10 - (46/7)
3x = (70 - 46)/7
3x = 24/7
x = 8/7
Step 5: The solution is (8/7, 23/7).
I hope these explanations and examples have helped clarify the steps involved in solving systems of equations using substitution and substitution with a variable. Remember, practice is key to mastering these methods!