What Is Linear Equations With Distribution?
A linear equation with distribution has brackets that must be multiplied out first. Distribute each outside number across its bracket, combine like terms, collect the x terms on one side and solve.
Enter the six numbers of a(bx + c) + d = e(fx + g) + h. The calculator expands both sides, simplifies, solves for x and substitutes the answer back to confirm it.
It also handles the special cases. If the x terms cancel and the constants disagree, there is no solution. If both sides become identical, every x is a solution.
How to Use the Linear Equations With Distribution Calculator
- Enter the eight numbers.
- Read each step.
- Check the solution.
- See the two lines cross on the chart.
Linear Equations With Distribution Formula
Distributive property
Linear Equations With Distribution Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| a, e | Numbers outside the brackets | |
| b, f | Coefficients of x inside | |
| c, g | Constants inside the brackets | |
| d, h | Constants outside |
Linear Equations With Distribution Worked Example
Solve 3(2x − 5) + 4 = 2(x + 3) − 1. Distribute: 6x − 15 + 4 = 2x + 6 − 1. Combine: 6x − 11 = 2x + 5. Subtract 2x: 4x − 11 = 5, add 11: 4x = 16, so x = 4. Check: 3(3) + 4 = 13 and 2(7) − 1 = 13.
With the example values loaded in the calculator above, the steps are:
- Equation: 3(2x − 5) + 4 = 2(x + 3) − 1.
- Distribute: 6x − 15 + 4 = 2x + 6 − 1.
- Combine like terms on each side: 6x − 11 = 2x + 5.
- Move the x terms to the left and the numbers to the right: 4x = 16.
- Divide both sides by 4: x = 16 ÷ 4 = 4.
- Check: left side = 13, right side = 13.
Result: Solution = x = 4.
How to Read the Solution
Most such equations have exactly one solution. For 3(2x − 5) + 4 = 2(x + 3) − 1, the sides become 6x − 11 and 2x + 5, which gives 4x = 16 and x = 4. Substituting back gives 13 on both sides.
Two special cases exist. If the x terms cancel and the numbers disagree, such as 2x + 6 = 2x + 7, there is no solution. If both sides become identical, every value of x works. The calculator labels these cases.
Where Linear Equations With Distribution Is Used
Homework
Check each step from distributing to checking the answer.
Test practice
Work through multi-step equations with brackets on both sides.
Tutoring
Show a student where each term comes from.
Spotting special cases
See how an equation can have no solution or infinitely many.
Common Linear Equations With Distribution Mistakes to Avoid
- Distribute first, then combine like terms, then collect x terms on one side.
- A negative outside a bracket changes the sign of every term inside it.
- Check your answer by putting it back into the original equation, not the simplified one.
- If the x terms vanish, look at the leftover numbers to decide between no solution and every solution.
Linear Equations With Distribution Terms and Definitions
- Identity
- An equation true for every value of x.
Linear Equations With Distribution FAQs
How do you solve equations with the distributive property?
Multiply out each bracket, combine like terms on each side, move the x terms to one side and the numbers to the other, then divide by the coefficient of x.
What if the x terms cancel?
Then compare what is left. If it is false, such as 6 = 7, there is no solution. If it is true, such as 6 = 6, every value of x is a solution.
How do I distribute a negative?
Change the sign of every term inside the bracket: −(x − 4) = −x + 4. A common slip is changing only the first term.
How can I check my answer?
Put the value of x back into the original equation and compute both sides. For x = 4 in the example, the left side is 13 and the right side is 13, so the answer is correct.