Solve for X Calculator

Q: What if the equation has no solution?

If the variable cancels and a false statement like 1 = 2 remains, the calculator reports that there is no solution.

About the Solve for X Calculator

To solve for $x$ means to find every value of $x$ that makes the equation true. For a linear equation that is done by undoing operations in reverse order; for higher powers it takes factoring or a formula.

The calculator chooses the method from the structure of your equation and substitutes every answer back into the original to confirm it. Values that would make a denominator zero are rejected.

What does solving for x mean?

An equation is a statement that two expressions are equal, and the letter $x$ stands for a number you do not yet know. Solving for $x$ means finding every number that makes the statement true, called the solution set. For $3(x - 2) = 2x + 7$ there is exactly one such number, 13, because both sides equal 33 when $x = 13$ and no other value balances them.

The guiding idea is balance: whatever you do to one side you must do to the other, so the equation stays true while it gets simpler. Each step undoes one operation that is acting on $x$ , working from the outside in, until $x$ stands alone. Some equations have one solution, some have several, some have none, and some are true for every value of $x$ .

How to use the Solve for X Calculator

Type the equation with one equals sign, such as 3(x − 2) = 2x + 7.
Optionally enter another variable to solve for, for example b in A = (1/2)·b·h.
Press Solve and read the steps from top to bottom.

Formulas

Inverse operations

x + a = b \Rightarrow x = b - a,\qquad ax = b \Rightarrow x = \frac{b}{a}

Linear form

ax + b = 0 \Rightarrow x = -\frac{b}{a},\ a \neq 0

Quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Worked example: 3(x − 2) = 2x + 7

The worked example solves $3(x - 2) = 2x + 7$ . The calculator distributes the 3 to get $3x - 6 = 2x + 7$ , subtracts $2x$ from both sides to collect the variable on the left, leaving $x - 6 = 7$ , and adds 6 to both sides to reach $x = 13$ . The check substitutes 13 into the original equation: the left side is $3 \times 11 = 33$ and the right side is $26 + 7 = 33$ .

The solution below is generated by the same engine that powers the calculator above.

Final answer

x = 13

How we read your problem

A linear equation in x: isolate x using inverse operations.

3 \left( x-2\right) = 2x+7

Step-by-step solution

Start with the equation
$3 \left( x-2\right) = 2x+7$
Simplify each side (distribute and combine like terms)
$3x - 6 = 2x + 7$
Subtract 2x from both sides
$x - 6 = 7$
Add 6 to both sides
$x = 13$

Formulas used

Linear equation
$ax + b = c \;\Rightarrow\; x = \frac{c - b}{a}$

Doing the same operation to both sides keeps the equation balanced. Undoing addition, then multiplication, leaves x alone.

Verification

x = 13: left side = 33, right side = 33 ✓

Detailed explanation

Equations with fractions

Fractions are cleared by multiplying every term by the least common denominator. In $\frac{x}{2} + \frac{x}{3} = 10$ the LCD is 6, which turns the equation into $3x + 2x = 60$ , so $5x = 60$ and $x = 12$ . Multiplying both sides by the same non-zero number never changes the solutions, which is why this step is safe. When the denominator contains $x$ the situation is different, because the number you multiply by could be zero for some values of $x$ .

Extraneous roots in rational equations

Take $\frac{x}{x - 2} = \frac{2}{x - 2} + 3$ . Clearing the denominators and simplifying leads to a quadratic whose only root is $x = 2$ . But $x = 2$ makes both denominators zero, so the original equation is not even defined there. The calculator notes the restriction when it multiplies, removes $x = 2$ at the end, and reports that the equation has no solution. Checking every candidate in the original equation is the only reliable way to catch roots like this.

No solution or infinitely many

Sometimes the variable disappears. $2(x + 1) = 2x + 5$ simplifies to $2x + 2 = 2x + 5$ , and subtracting $2x$ leaves $2 = 5$ , which is false whatever $x$ is, so there is no solution. If instead the leftover statement is true, such as $2 = 2$ , every real number works and the equation is an identity. Both outcomes are reported in words rather than as a value of $x$ .

Checking a solution by substitution

A solution is only correct if it makes the original equation true, so the calculator always substitutes its answer back into the equation you typed, not into a simplified version. Substituting into a later line would hide any mistake made before it. Do the same by hand: evaluate the left side and the right side separately, and if the two numbers differ, look for the first step where the two sides stopped balancing.

Common mistakes

Applying an operation to only one side of the equation.
Distributing a negative sign to the first term in brackets only.
Dividing both sides by $x$ , which loses the solution $x = 0$ .
Keeping a root that makes a denominator zero in the original equation.
Moving a term across the equals sign without changing its sign.

Tips

Clear fractions first by multiplying both sides by the least common denominator.
Collect variable terms on the side where their coefficient stays positive.
Always substitute your answer back into the original equation.
Write each step on a new line with the equals signs lined up; it makes sign errors easy to spot.
Use the Solve for box to rearrange a formula, such as finding $b$ in $A = \frac12 bh$ .

Frequently asked questions

How do I solve for x?

Simplify each side, move all x-terms to one side and constants to the other using inverse operations, then divide by the coefficient of x. For $3(x-2) = 2x + 7$ : $3x - 6 = 2x + 7$ , so $x = 13$ .

Can it solve for a variable other than x?

Yes. Enter the letter in the Solve for box. Formulas such as $A = \frac12 bh$ are rearranged to $b = \frac{2A}{h}$ when the variable appears linearly.

What if the equation has no solution?

If the variable cancels and a false statement like $1 = 2$ remains, the calculator reports that there is no solution.

What if every x works?

If the variable cancels and a true statement remains, the equation is an identity and all real numbers are solutions.

Does it handle x squared?

Yes. Quadratic equations are solved exactly with factoring or the quadratic formula, including irrational and complex roots.

Why was one of my solutions removed?

When an equation has x in a denominator, multiplying through can create extraneous roots. Any root that makes a denominator zero is discarded and the reason is shown.

How do I solve an equation with fractions?

Multiply every term on both sides by the least common denominator. For $\frac{x}{2} + \frac{x}{3} = 10$ , multiplying by 6 gives $3x + 2x = 60$ , so $x = 12$ .

How do I rearrange a formula for one variable?

Treat the other letters as fixed numbers and undo the operations around the target variable. From $A = \frac12 bh$ , multiply by 2 and divide by $h$ to get $b = \frac{2A}{h}$ .

Solve your equation for x

Type your equation above and get every solution with the inverse operation behind each step and a substitution check.

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