4x+10y=28
6x+7y=6
Let’s eliminate x:
4x + 10y = 28, so multiply both sides by 50% to get:
6x + 15y = 42, but we know that:
6x + 7y = 6, so subtract one from the other to get:
8y = 42 - 6 = 36, so
y = 36/8 = 4.5, so
10y = 45, but
4x + 10y = 4x + 45 = 28, so
4x = 28 - 45 = -17, so
x = -17/4 = -4.25
Check it:
4x + 10y = -17 + 45 = 28 Correct
6x + 7y = -25.5 + 31.5 = 6 Correct.
So x = -4.25 and y = 4.5
To solve this system of equations, you can use the method of substitution or the method of elimination. I will explain both methods and then demonstrate each one using your specific equations.
Method 1: Substitution
1. Solve one of the equations for one variable in terms of the other variable.
2. Substitute this expression into the other equation.
3. Solve the resulting equation for one variable.
4. Substitute this value back into one of the original equations to find the value of the other variable.
5. Check the solution by substituting the values of the variables into both equations.
Method 2: Elimination
1. Multiply one or both of the equations by a constant(s) in order to create coefficients that are additive inverses (i.e., sum to zero) for one variable.
2. Add or subtract the equations to eliminate one variable.
3. Solve the resulting equation for the remaining variable.
4. Substitute this value back into one of the original equations to find the value of the other variable.
5. Check the solution by substituting the values of the variables into both equations.
Now, let's solve the given system of equations (4x + 10y = 28, 6x + 7y = 6) using both methods.
Method 1: Substitution
1. From the first equation, solve for x:
4x = 28 - 10y
x = (28 - 10y) / 4
2. Substitute this expression for x into the second equation:
6((28 - 10y) / 4) + 7y = 6
3. Simplify the equation:
(168 - 60y) / 4 + 7y = 6
(168 - 60y + 28y) / 4 = 6
(168 - 32y) / 4 = 6
4. Multiply both sides by 4 to get rid of the fraction:
168 - 32y = 24
5. Solve for y:
-32y = 24 - 168
-32y = -144
y = -144 / -32
y = 4.5
6. Substitute the value of y back into one of the original equations. Let's use the first equation:
4x + 10(4.5) = 28
4x + 45 = 28
4x = 28 - 45
4x = -17
x = -17 / 4
x = -4.25
Therefore, the solution to the system of equations is x = -4.25 and y = 4.5. You can verify this solution by substituting these values back into the original equations.
Method 2: Elimination
1. Multiply the first equation by 6 and the second equation by 4 to create coefficients that are additive inverses for x:
24x + 60y = 168
24x + 28y = 24
2. Subtract the second equation from the first equation:
(24x + 60y) - (24x + 28y) = 168 - 24
32y = 144
y = 144 / 32
y = 4.5
3. Substitute the value of y back into one of the original equations. Let's use the first equation:
4x + 10(4.5) = 28
4x + 45 = 28
4x = 28 - 45
4x = -17
x = -17 / 4
x = -4.25
Again, the solution to the system of equations is x = -4.25 and y = 4.5.