How to solve the simultaneous equations 3x + y = 17 and y = x + 1 ?
There are a few ways to do this. In this case the easiest is probably substituting y in the first equation for x+1
so
3x + (x + 1) =17
remove the brackets
3x + x + 1 = 17
combine the x terms
4x + 1 = 17
move the 1 from the LHS to the RHS
4x +1 -1 = 17 - 1
4x = 16
divide both sides by 4
4x/4 = 16/4
x=4
as
y=x+1
y= 4 + 1
y=5
To solve the simultaneous equations 3x + y = 17 and y = x + 1, you can use the method of substitution or the method of elimination. Let's go through both methods:
Method 1: Substitution
Step 1: Solve one equation for one variable.
We have the equation y = x + 1. Let's solve it for y.
Rearranging the equation, we have x = y - 1.
Step 2: Substitute the expression into the other equation.
Now, substitute the expression for x (from Step 1) into the other equation (3x + y = 17).
Replace x with y - 1, giving us:
3(y - 1) + y = 17
Step 3: Simplify and solve for y.
Distribute the 3:
3y - 3 + y = 17
Combine like terms:
4y - 3 = 17
Add 3 to both sides:
4y = 20
Divide by 4:
y = 5
Step 4: Substitute the value of y back into either of the original equations and solve for x.
Using y = 5, substitute it into y = x + 1:
5 = x + 1
Subtract 1 from both sides:
x = 4
Therefore, the solution to the simultaneous equations 3x + y = 17 and y = x + 1 is x = 4 and y = 5.
Method 2: Elimination
Step 1: Multiply one or both equations by suitable numbers to make the coefficients of one variable the same or multiples of each other.
Both equations are already in a form where the coefficients of y are the same, so we can directly proceed to the next step.
Step 2: Add or subtract the equations to eliminate one of the variables.
Since the coefficients of y in both equations are 1, we can simply subtract the equations.
(3x + y) - (y) = (17) - (x + 1)
3x - x = 17 - 1
2x = 16
Divide by 2:
x = 8
Step 3: Substitute the value of x back into either of the original equations and solve for y.
Using x = 8, substitute it into 3x + y = 17:
3(8) + y = 17
24 + y = 17
Subtract 24 from both sides:
y = 17 - 24
y = -7
Therefore, the solution to the simultaneous equations 3x + y = 17 and y = x + 1 is x = 8 and y = -7.
Both methods should give you the same solution, which is x = 4 and y = 5.