Solve The Following System by Substitution Calculator

How to solve a system by substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. Here's how it works:

1. Start with a system of two equations with two variables
2. Solve one equation for one variable in terms of the other
3. Substitute this expression into the second equation
4. Solve the resulting equation for the remaining variable
5. Find the value of the first variable using the expression from step 2
6. Write the solution as an ordered pair (x, y)

Step-by-step substitution method

Let's solve the general system:

1. Choose one equation to solve for one variable. For example, solve equation 1 for x:
   a₁x = c₁ - b₁y x = (c₁ - b₁y)/a₁
2. Substitute this expression into equation 2:
   a₂[(c₁ - b₁y)/a₁] + b₂y = c₂
3. Multiply through by a₁ to eliminate the denominator:
   a₂(c₁ - b₁y) + a₁b₂y = a₁c₂
4. Distribute and combine like terms:
   a₂c₁ - a₂b₁y + a₁b₂y = a₁c₂ (a₂b₁ - a₁b₂)y = a₁c₂ - a₂c₁
5. Solve for y:
   y = (a₁c₂ - a₂c₁)/(a₂b₁ - a₁b₂)
6. Substitute y back into the expression for x:
   x = (c₁ - b₁[(a₁c₂ - a₂c₁)/(a₂b₁ - a₁b₂)])/a₁

The solution is the ordered pair (x, y).

Worked examples

Example 1: Simple system

Solve:

1. Solve equation 2 for y:
   y = 4x - 6
2. Substitute into equation 1:
   2x + 3(4x - 6) = 12 2x + 12x - 18 = 12 14x = 30 x = 30/14 = 15/7
3. Find y:
   y = 4(15/7) - 6 = 60/7 - 42/7 = 18/7
4. Solution: (15/7, 18/7)

Example 2: System with fractions

Solve:

1. Multiply both equations by 6 to eliminate denominators:
   3x + 2y = 30 2x - 3y = 6
2. Solve equation 1 for x:
   x = (30 - 2y)/3
3. Substitute into equation 2:
   2[(30 - 2y)/3] - 3y = 6 (60 - 4y)/3 - 3y = 6 20 - (4/3)y - 3y = 6 20 - (13/3)y = 6 -(13/3)y = -14 y = 42/13
4. Find x:
   x = (30 - 2(42/13))/3 = (390/13 - 84/13)/3 = (306/13)/3 = 102/13
5. Solution: (102/13, 42/13)