3. Solve the systems of equations by elimination
3x-5y = 29
2x + 3y = -6
To solve the system of equations by elimination, we can multiply the first equation by 2 and multiply the second equation by 3 to make the coefficient of x in both equations the same.
2(3x - 5y) = 2(29)
3(2x + 3y) = 3(-6)
Simplifying these equations, we get:
6x - 10y = 58
6x + 9y = -18
Now, we can subtract the second equation from the first equation to eliminate the x term:
(6x - 10y) - (6x + 9y) = 58 - (-18)
6x - 10y - 6x - 9y = 58 + 18
-19y = 76
y = 76 / -19
y = -4
Substituting this value of y back into either of the original equations, we can solve for x:
3x - 5(-4) = 29
3x + 20 = 29
3x = 9
x = 9 / 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -4.
To solve the system of equations by elimination, we want to eliminate one of the variables by adding or subtracting the equations together. Let's start with the given system:
1) 3x - 5y = 29
2) 2x + 3y = -6
To eliminate one of the variables, we need to multiply one or both equations by a constant so that when we add or subtract them, one of the variables will cancel out. In this case, we can eliminate the variable "x" by multiplying equation (1) by 2 and equation (2) by 3:
3) 6x - 10y = 58 (equation 1 multiplied by 2)
4) 6x + 9y = -18 (equation 2 multiplied by 3)
Now, we can subtract equation (4) from equation (3) in order to eliminate the variable "x":
(6x - 10y) - (6x + 9y) = (58) - (-18)
6x - 10y - 6x - 9y = 58 + 18
-19y = 76
Next, divide both sides of the equation by -19 to solve for "y":
-19y / -19 = 76 / -19
y = -4
Now that we have the value of "y", we can substitute it back into one of the original equations (either equation 1 or equation 2) to solve for "x". Let's use equation (1):
3x - 5(-4) = 29
3x + 20 = 29
3x = 29 - 20
3x = 9
x = 9 / 3
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -4.
To solve the system of equations by elimination, we need to eliminate one variable by adding or subtracting the equations.
Step 1: Multiply the first equation by 2 and the second equation by 3 to make the coefficients of x in both equations equal.
Equation 1: 3x - 5y = 29
Multiply by 2: 2(3x - 5y) = 2(29)
Simplifying: 6x - 10y = 58
Equation 2: 2x + 3y = -6
Multiply by 3: 3(2x + 3y) = 3(-6)
Simplifying: 6x + 9y = -18
Step 2: Now, we can add the two equations to eliminate x.
(6x - 10y) + (6x + 9y) = 58 + (-18)
Simplifying: 12x - y = 40
Step 3: We now have a single equation in terms of y. Let's solve for y.
12x - y = 40
Isolate y: -y = 40 - 12x
Multiply by -1 to switch the sign: y = -40 + 12x
Step 4: Substitute the value of y back into one of the original equations to solve for x.
From the first original equation:
3x - 5y = 29
Substituting y = -40 + 12x:
3x - 5(-40 + 12x) = 29
Distribute -5 to both terms in the parentheses:
3x + 200 - 60x = 29
Combine like terms:
-57x + 200 = 29
Subtract 200 from both sides:
-57x = -171
Divide by -57:
x = 3
Step 5: Now that we know the value of x, we can substitute it back into the equation we found for y to solve for y.
y = -40 + 12x
y = -40 + 12(3)
y = -40 + 36
y = -4
Therefore, the solution to the system of equations is x = 3 and y = -4.