i need help with these problems (either a solution or how to plug them into a graphing calculator):
1) Solve by substitution or elimination
4/x + 1/y + 2/z = 4
2/x + 3/y - 1/z = 1
1/x + 1/y + 1/z = 4
2)Solve the system of equations by substitution.
x^2 + y^2 = 25
x - y = 7
3) Solve the system of equations by substitution. Round the answer to the nearest hundredths.
9x^2 + 16y^2 = 140
x^2 - 4y^2 = 4
4)Solve by elimination:
1/x + 3/y = 7
4/x - 2/y = 1
Thanks!
1. add the 2nd and 3rd
----> 3/x + 4/y = 5 (#4)
double the 2nd and add to the 1st
---> 8/x + 7/y = 6 (#5)
8 times #4 --> 24/x + 32/y = 40
3 times #5 --> 24/x + 21/y = 18
subtract:
11/y = 22
11 = 22y
y = 1/2
sub back into either #4 or #4 to get x
then back into 1st to get z
2. from the second: x = y+7
into the first: (y+7)^2 + y^2 = 25
2y^2 + 14y + 24 = 0
y^2 + 7y + 12 = 0
(y+3)(y+4) = 0
y = -3 or y = -4
sub both of those into x = y+7 to get their corresponding x values
3. from 2nd ---> x^2 = 4y^2 + 4
sub into 1st, etc
4. easier than #1
multiply 1st by 4, then subtract the 2nd
what? im confused
Okay;
let,(x-y)^2=x^2+y^2-2xy.
But,x-y=7 and x^2+y^2=25.by apply the formula.
(7)^2=25-2xy
49=25-2xy
xy=-12. Then also,
(x+y)^2=(x-y)^2+4xy
(x+y)^2=49-48.
X+y=+/-1. Then solve the eqn.
X+y=+/-1
x-y=7. Then
x=3 or 4.
Y=-3 or -4.
Sure, I can help you with these problems. Let's go through each problem one by one and explain how to solve them.
Problem 1: Solve by substitution or elimination
To solve this system of equations, we'll use the method of elimination.
Step 1: First, let's eliminate the z terms. Multiply the second equation by 2 and the third equation by 1 to balance the coefficients of z:
2(2/x + 3/y - 1/z = 1) gives us 4/x + 6/y - 2/z = 2
1(1/x + 1/y + 1/z = 4) gives us 1/x + 1/y + 1/z = 4
Step 2: Subtract the second equation from the first equation:
(4/x + 1/y + 2/z) - (4/x + 6/y - 2/z) = 4 - 2
Simplifying, we get: -5/y + 4/z = 2
Step 3: Subtract the third equation from the first equation:
(4/x + 1/y + 2/z) - (1/x + 1/y + 1/z) = 4 - 4
Simplifying, we get: 3/x + 1/z = 0
Step 4: Now we have a system of two equations:
-5/y + 4/z = 2
3/x + 1/z = 0
You can solve this system of equations using either substitution or elimination. Since we already used elimination, let's use substitution to find the solution.
We will solve the second equation for x:
3/x + 1/z = 0
Rearranging, we get:
3/x = -1/z
Cross-multiplying, we get:
-x = 3z
Solving for x, we get:
x = -3z
Now substitute x = -3z into the first equation:
-5/y + 4/z = 2
Substituting x = -3z, we get:
-5/y + 4/z = 2
Now we have a system of two equations:
-5/y + 4/z = 2
x = -3z
You can solve this new system using either substitution or elimination to find the values of x, y, and z.
Problem 2: Solve the system of equations by substitution
To solve this system of equations, we will use the substitution method.
Step 1: From the second equation, solve for x in terms of y:
x - y = 7
x = y + 7
Step 2: Substitute x = y + 7 into the first equation:
(y + 7)^2 + y^2 = 25
This is a quadratic equation in y. Simplify and solve for y by using factoring or the quadratic formula. Once you find the value of y, substitute it back into the equation x = y + 7 to find x.
Problem 3: Solve the system of equations by substitution. Round the answer to the nearest hundredths.
To solve this system, we will use the substitution method.
Step 1: Solve the second equation for x in terms of y:
x^2 - 4y^2 = 4
x^2 = 4y^2 + 4
x = sqrt(4y^2 + 4) or x = -sqrt(4y^2 + 4)
Step 2: Substitute these values of x into the first equation:
9x^2 + 16y^2 = 140
9(4y^2 + 4) + 16y^2 = 140
This is a quadratic equation in y. Simplify and solve for y by using factoring or the quadratic formula. Once you find the value(s) of y, substitute it back into the equation x = sqrt(4y^2 + 4) or x = -sqrt(4y^2 + 4) to find x.
Problem 4: Solve by elimination
To solve this system of equations, we will use the method of elimination.
Step 1: Multiply the first equation by 2 and the second equation by 1 to balance the coefficients of y:
2(1/x + 3/y) = 2(7) gives us 2/x + 6/y = 14
1(4/x - 2/y) = 1(1) gives us 4/x - 2/y = 1
Step 2: Add the equations together:
(2/x + 6/y) + (4/x - 2/y) = 14 + 1
Simplifying, we get: 6/x = 15
Step 3: Solve for x:
6/x = 15
Cross multiplying, we get: 15x = 6
Dividing by 15, we get: x = 6/15 or x = 2/5
Now substitute x = 2/5 into either equation to find the value of y, and you will have the solution to the system of equations.
These are the steps to solve each of the systems of equations. If you have a graphing calculator, you can also plot the equations and find the intersection points to get the solution.