How do I solve the following by substitution:
5x -2y =-5
y - 5x = 3
8x - 4y = 16
y = 2x - 4
4x -12y =5
-x + 3y = -1
5x -2y =-5
y - 5x = 3
5x -2y =-5
-5x +y = 3
5x -2y =-5
10x -2y =-6
15x =-1
x= -1/15
(-1/15x5) -2y =-5
-1/3 -2y=-5
+1/3 +1/3
-2y= 4 2/3
/-2 /-2
y= -2 1/3
x= -1/15
y= -2 1/3
Thank you for your help!
lol np bro
To solve a system of equations by substitution, you need to follow these steps:
1. Choose one equation and solve it for one variable in terms of the other variable.
2. Substitute the expression you found in step 1 into the other equation.
3. Solve the resulting equation for the remaining variable.
4. Substitute the value obtained in step 3 back into either of the original equations to find the value of the other variable.
5. Finally, substitute the values of both variables into any of the original equations to check if they satisfy the equations.
Now let's solve the given systems of equations using the substitution method.
System 1:
Equation 1: 5x - 2y = -5
Equation 2: y - 5x = 3
Step 1: Solve Equation 2 for y in terms of x.
y = 5x + 3
Step 2: Substitute the expression for y from Step 1 into Equation 1.
5x - 2(5x + 3) = -5
Step 3: Simplify and solve for x.
5x - 10x - 6 = -5
-5x - 6 = -5
-5x = -5 + 6
-5x = 1
x = 1 / -5
x = -1/5
Step 4: Substitute the value of x into Equation 2 to find y.
y - 5(-1/5) = 3
y + 1 = 3
y = 3 - 1
y = 2
Step 5: Substitute the values of x and y into any of the original equations to check the solution.
Using Equation 1: 5(-1/5) - 2(2) = -5
-1 - 4 = -5 (This is true)
Therefore, the solution for System 1 is x = -1/5 and y = 2.
Now, let's solve the other systems of equations using the same method.
System 2:
Equation 1: 8x - 4y = 16
Equation 2: y = 2x - 4
Step 1: Since equation 2 is already solved for y in terms of x, we can skip this step.
Step 2: Substitute the expression for y from Step 1 into Equation 1.
8x - 4(2x - 4) = 16
Step 3: Simplify and solve for x.
8x - 8x + 16 = 16
16 = 16 (This equation is true)
This means that both equations are the same and infinite solutions exist.
Therefore, for System 2, there are infinite solutions.
System 3:
Equation 1: 4x - 12y = 5
Equation 2: -x + 3y = -1
Step 1: Solve Equation 2 for x in terms of y.
-x = -3y - 1
x = 3y + 1
Step 2: Substitute the expression for x from Step 1 into Equation 1.
4(3y + 1) - 12y = 5
Step 3: Simplify and solve for y.
12y + 4 - 12y = 5
4 = 5 (This equation is not true)
This means that there is no solution for this system of equations.
Therefore, for System 3, there is no solution.
To summarize:
System 1: x = -1/5, y = 2 (has a unique solution)
System 2: Infinite solutions
System 3: No solution