solve the equation: First: x + 3y = 5 Second: 3x - y = 5
should i use substitution or add them?
"First" + 3 times "Second" eliminates y:
x + 3y + 3*(3x - y) = 5 + 3*5 --->
10 x = 20 ---->
x = 2
Inserting in "First" gives:
2 + 3 y = 5 --->
3 y = 3 ---->
y = 1
This problem is a bit too easy to practice solving equations. You should try to practice yourself by giving three variables x1, x2, x3 arbitrary values, say
x1 = 4, x2= 13, x3 = 3
and then evaluate three different linear functions of the x1, x2 and x3, e.g.
x1 + 2 x2 - 7 x3 = 9
2 x1 - x2 + x3 = 6
-x1 + 5 x2 - 3 x3 = 52
Then try to solve these three equations. If you do it correctly you should obtain the values you used for x1 x2 and x3 for the solution. It can happen, though, that you find more solutions than the values you started out with.
To solve the given system of equations, you can use either the substitution method or the addition/elimination method. Here's how to use both methods:
1. Substitution method:
- Solve one equation for one variable in terms of the other variable.
- From the first equation, solve for x: x = 5 - 3y.
- Substitute the expression for x into the other equation.
- In the second equation, replace x with (5 - 3y): 3(5 - 3y) - y = 5.
- Simplify the equation: 15 - 9y - y = 5.
- Combine like terms: -10y = -10.
- Solve for y: y = -10 / -10 = 1.
- Substitute the value of y back into either of the original equations to find x.
- Using the first equation: x + 3(1) = 5.
- Simplify: x + 3 = 5.
- Solve for x: x = 2.
Therefore, the solution to the system of equations is x = 2 and y = 1.
2. Addition/Elimination method:
- Multiply both equations by appropriate constants to make the coefficients of one variable opposite in sign.
- In this case, you can multiply the first equation by 3 to cancel out the x term when added to the second equation.
- First equation: 3(x + 3y) = 3(5) becomes 3x + 9y = 15.
- Second equation remains the same: 3x - y = 5.
- Add the two equations together to eliminate one variable.
- (3x + 9y) + (3x - y) = 15 + 5.
- Simplify the equation: 6x + 8y = 20.
- Solve for one variable in terms of the other.
- You can solve for x: 6x = 20 - 8y.
- Divide both sides by 6: x = (20 - 8y) / 6 = (10 - 4y) / 3.
- Substitute the expression for x back into either of the original equations to find y.
- Using the first equation: (10 - 4y) / 3 + 3y = 5.
- Multiply through by 3 to get rid of the denominator: 10 - 4y + 9y = 15.
- Simplify and solve for y: 5y = 5, y = 1.
- Substitute the value of y back into the expression for x to find x.
- Using x = (10 - 4y) / 3: x = (10 - 4(1)) / 3 = 2.
Again, the solution is x = 2 and y = 1.
Both methods yield the same solution, so you can choose whichever method you find more intuitive or convenient for you.