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To combine like terms, add or subtract the coefficients of terms that have exactly the same variable part and leave that variable part unchanged. For example, (5x + 3x = 8x), but (5x + 3y) cannot be combined because (x) and (y) are different variables. This works for any polynomial expression, with one extra step: if parentheses are present, distribute first, then combine.

What Counts as a "Like Term"

Two terms are like terms only when their variable parts match completely — same variables, same exponents. The coefficients do not need to match; in fact, they are the numbers you combine.

A useful habit: rewrite each term so variables appear in alphabetical order ((2yx \to 2xy)) before deciding whether terms match.

The Core Procedure

1. Scan the expression and mark terms with identical variable parts.
2. Keep the sign in front of each term. In (8x - 3x + 2x), the terms are (8x), (-3x), and (2x).
3. Add the coefficients of each group: (8 - 3 + 2 = 7), so the (x)-terms become (7x).
4. Write the simplified expression with the combined terms and any terms that had no partner.

Worked example:

[ 9m + 4 - 2m + 7 ]

• (m)-terms: (9m - 2m = 7m)
• Constants: (4 + 7 = 11)
• Result: (7m + 11)

Handling Negative Signs and Subtraction

Most errors come from dropping a minus sign. Treat subtraction as adding a negative term, and move the sign with the term when you regroup:

[ -6k + 10 - k - 15 = (-6k - k) + (10 - 15) = -7k - 5 ]

If a term has no visible coefficient, it is (1): (-k) means (-1k), and (x) means (1x).

When Parentheses Are Involved: Distribute First

You cannot combine terms across a set of parentheses until the parentheses are removed. Apply the distributive property, then combine.

[ 3(2x + 5) - 4(x - 1) ]

1. Distribute: (6x + 15 - 4x + 4) (note: (-4 \times -1 = +4))
2. Combine: ((6x - 4x) + (15 + 4) = 2x + 19)

Watch the sign on the second distribution — a negative multiplier flips the sign of every term inside the parentheses.

Verify Your Answer

Substitute a simple number for the variable in both the original and simplified expressions; the values should match.

Using (x = 2) in the example above:

• Original: (3(4 + 5) - 4(2 - 1) = 27 - 4 = 23)
• Simplified: (2(2) + 19 = 23)

A match does not prove the simplification is correct for every value, but a mismatch reliably signals an error — usually a sign or a distribution mistake.

Common Pitfalls

• Combining unlike terms. (3x^2 + 4x) stays as is; it does not become (7x^2) or (7x).
• Changing the exponent. Combining (2x^3 + 5x^3) gives (7x^3), not (7x^6). Exponents are never added when combining terms.
• Losing signs during regrouping. Keep each sign attached to its term, especially when moving terms around.
• Skipping distribution. Parentheses must be cleared before any combining happens.

Where to Practice

Algebra House (algebrahouse.com) offers algebra examples, answered questions, and state test and ACT/SAT practice sections, which is a reasonable place to find additional worked problems and check your steps against a full solution.

To write the equation of a line, first identify what you are given, then choose the matching form: slope-intercept form (y = mx + b) when you know the slope and y-intercept, point-slope form (y - y_1 = m(x - x_1)) when you know a point and the slope, and standard/general form (Ax + By = C) when you need a tidy equation with integer coefficients. Two points or a graph always let you find the slope first, after which either slope-intercept or point-slope form finishes the job. The sections below give the steps, a worked example for each case, and a quick way to check your answer.

The three forms and when to use each

All three describe the same line, so converting between them is just algebra. Slope-intercept form is usually the most readable for graphing; point-slope form is the fastest route from a point and a slope; standard form is common in textbooks and when solving systems of equations.

Writing the equation from a slope and y-intercept

If you are told the slope is (m) and the line crosses the y-axis at ((0, b)), substitute directly into (y = mx + b).

Example: slope (3), y-intercept (-2) gives (y = 3x - 2).

Check: substitute (x = 0); you should get (y = -2), confirming the intercept.

Writing the equation from a point and a slope

Use point-slope form, then simplify to slope-intercept form if requested.

Example: slope (-2) through ((4, 1)).

1. Substitute: (y - 1 = -2(x - 4)).
2. Distribute: (y - 1 = -2x + 8).
3. Add 1: (y = -2x + 9).

Check: plug in (x = 4): (y = -2(4) + 9 = 1), which matches the given point.

Writing the equation from two points

Two points determine the slope; then use either point in point-slope form.

Example: through ((1, 5)) and ((3, 11)).

1. Slope: (m = \dfrac{11 - 5}{3 - 1} = \dfrac{6}{2} = 3).
2. Point-slope with ((1, 5)): (y - 5 = 3(x - 1)).
3. Simplify: (y = 3x + 2).

Check: the second point should satisfy the equation: (3(3) + 2 = 11). ✓

A useful special case: a horizontal line through points with the same y-value has slope (0), so its equation is (y = c). A vertical line through points with the same x-value has an undefined slope and equation (x = c); it cannot be written in slope-intercept form.

Reading the equation from a graph

1. Find the y-intercept: the point where the line crosses the y-axis gives (b).
2. Find the slope: pick two clear lattice points and compute (\dfrac{\text{rise}}{\text{run}}).
3. Substitute into (y = mx + b).

Example: a line crossing the y-axis at ((0, -1)) and passing through ((2, 3)) has slope (\dfrac{3 - (-1)}{2 - 0} = 2), so (y = 2x - 1).

Check: confirm a third point on the graph satisfies the equation before you commit to it.

Verifying your equation

• Substitute known points. Every point you were given must make the equation true.
• Recompute the slope from your final equation and compare it with the slope you found.
• Check the intercept by setting (x = 0) (for slope-intercept form) and comparing with the graph or given data.

If a point fails, the most common causes are a sign error when subtracting coordinates in the slope formula, or distributing the slope to only one term in point-slope form.

Common sticking points

• Mixing up (x_1) and (y_1) in point-slope form — keep the point's coordinates in the same order as the labels.
• Dropping the negative sign when the slope is negative and you distribute.
• Treating the x-intercept as (b) — (b) is always the y-intercept.
• Forgetting vertical lines have no slope-intercept form.

For related practice, Algebra House (algebrahouse.com) offers worked algebra examples, answered questions, and state test and ACT/SAT practice sections that cover linear equations alongside topics such as combining like terms and the distributive property.

Tiger Algebra is a free online algebra solver and calculator that shows step-by-step solutions without requiring a download or signup. It runs in a mobile or desktop browser, so you can use it to check homework answers, study the steps behind a solution, or quickly work through common algebra problems. You enter an equation or problem, press Solve, and the site returns a solution with steps.

What Tiger Algebra covers

The homepage lists a broad set of solvers and topics. Based on the site's own listing, these include:

• Order of operations — e.g. 2+3*4, (8+4)/3
• Least common multiple (LCM) — e.g. lcm(12,20), lcm(18,24,30)
• Greatest common factor (GCF) — e.g. gcf(12,18), gcf(4,8,32)
• Linear equations with one unknown — e.g. 2x=8, 3x+2=7x-5
• Percentages — e.g. 20% of 80, 15 is what % of 60
• Factorials — e.g. 5!, 8!/5!
• Quadratic equations using the quadratic formula
• Linear equations with two unknowns
• Polynomial root calculator
• Scientific notation
• Simplifying radicals
• Factoring trinomials
• Polynomial long division
• Reducing fractions to lowest terms
• Arithmetic sequences
• Trigonometry — exact values, unit circle, and graphs

The site also shows a running count of problems solved (over 20 million at the time of the source page) and a "Latest Solved" feed that includes inequalities, distance and midpoint problems, and limit-style expressions such as slim(x→0)(sin(x)).

How to use it

1. Enter your problem. Type an equation or expression into the input field on the calculator page — for example 2x=8 or lcm(12,20).
2. Press Solve. The site processes the input and returns the answer with step-by-step solutions.
3. Use Learn, Practice, or Quiz. Each topic row on the homepage links to these modes, so you can read an explanation, work through practice problems, or test yourself.
4. Try camera input if needed. A camera input option is present, but the page notes that camera input is not recognized, so typing is the reliable path.

Expected result: a solved answer plus the steps used to reach it, which you can follow to understand the method rather than just the final number.

What it's good for — and its limits

Good fit:

• Checking your own work on standard algebra problems
• Seeing intermediate steps when you're stuck
• Quick calculation of LCM, GCF, factorials, percentages, and similar operations
• Practicing with the Learn/Practice/Quiz structure

Keep in mind:

• The source page does not state pricing, so free use is described by the site's own "free" label; no payment details are given.
• Camera input is listed but flagged as not recognized, so don't rely on it.
• The solver list covers the topics shown above; for problems outside that set, results may be limited.

If you want a no-signup way to get step-by-step algebra help in a browser, Tiger Algebra fits that use. Start by typing your equation and pressing Solve, then use the Learn and Practice links to go deeper on any topic you need.

CSS3 transform functions change how an element is drawn — moving, rotating, scaling, or skewing it — without affecting the layout of surrounding elements. You apply them through the transform property, and they work in both 2D and 3D space. Use them when you want visual effects like hover animations, card flips, or subtle motion; avoid them when you actually need to change an element's position in the document flow, since transforms are purely visual.

The Core 2D Functions

Each function takes one or more arguments and returns a transformed version of the element.

A simple hover effect:

.card {
  transition: transform 0.3s ease;
}
.card:hover {
  transform: scale(1.05) rotate(2deg);
}

The element grows slightly and tilts on hover. Because transforms don't trigger layout recalculation, this stays smooth.

3D Functions and Perspective

3D transforms add depth. The key functions are rotateX(), rotateY(), rotateZ(), and translate3d(). But rotation alone looks flat unless you add perspective.

perspective defines how far the viewer is from the Z=0 plane. A smaller value creates a more dramatic 3D effect:

.scene {
  perspective: 800px;
}
.box {
  transform: rotateY(45deg);
}

Without perspective, rotateY(45deg) just squashes the element horizontally. With it, the element appears to turn in space, with one edge closer to the viewer.

You can also apply perspective per-element with perspective() inside the transform:

.box {
  transform: perspective(600px) rotateX(30deg);
}

The visualizer at css-transform.moro.es lets you drag sliders for rotation, perspective, and scaling and see the result immediately — useful for finding values before committing them to code.

Transform Origin: Changing the Pivot

By default, transforms pivot around the element's center (50% 50%). transform-origin moves that pivot:

.box {
  transform-origin: top left;
  transform: rotate(30deg);
}

Now the element rotates around its top-left corner instead of its center. This is one of the most common sources of "why does it rotate like that?" confusion. If a rotation or scale looks wrong, check transform-origin first.

How Multiple Functions Combine

When you list several functions, they apply right to left — the last function in the list is applied first to the element's coordinate system.

transform: translateX(100px) rotate(45deg);

This rotates the element 45° first, then translates it 100px along the (now rotated) X axis. Swap the order and you get a different result:

transform: rotate(45deg) translateX(100px);

Here the element moves along the rotated axis, so it travels diagonally. Order matters, and it's the second most common gotcha after transform-origin.

Where to Experiment

The CSS Transform Functions Visualizer (css-transform.moro.es) is a playground for exactly this. Its interface covers Rotation (2D), Rotation (3D), Perspective, Transform Origin, and Scaling (Resizing), with front/back views for 3D. Adjust a value, watch the element respond, then copy the resulting transform into your stylesheet. This beats guessing numbers and refreshing a browser.

Practical Notes

• Transforms don't affect surrounding layout — the element's original space is preserved.
• They combine well with transition for animation, but for complex sequences use @keyframes.
• 3D transforms need perspective on the parent (or perspective() in the transform) to look three-dimensional.
• Always verify function order and transform-origin when a result surprises you.

An equation is a statement that two expressions are equal, and solving it means finding the value(s) of the unknown that make that statement true. For example, in x + 3 = 7, the solution is x = 4 because substituting 4 back gives 4 + 3 = 7. The method you use depends on the type of equation: linear equations need isolation of the variable, quadratics often need factoring or the quadratic formula, and systems need substitution or elimination. You can solve all of these by hand, or use a step-by-step solver like MathDF to check each transformation.

The core idea: a balance you must keep

An equation has two sides separated by "=". Whatever you do to one side, you must do to the other, or the equality breaks. This single rule generates every algebraic solving technique:

• Add or subtract the same quantity on both sides (used to move terms).
• Multiply or divide both sides by the same non-zero quantity (used to clear fractions or coefficients).
• Apply a function to both sides (squaring, taking roots) — but this can introduce extra solutions, so always verify.

The goal is to isolate the unknown on one side, ending with something like x = value.

Common types of equations and how to solve them

Linear equations (first degree)

Form: ax + b = 0, where a ≠ 0.

Steps:

1. Expand any brackets and combine like terms on each side.
2. Move all variable terms to one side and constants to the other.
3. Divide by the coefficient of the variable.

Example: 3(x − 2) = 9 → 3x − 6 = 9 → 3x = 15 → x = 5.

Quadratic equations (second degree)

Form: ax² + bx + c = 0, where a ≠ 0.

Three standard routes:

• Factoring when the expression splits neatly: x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0 → x = 2 or x = 3.
• Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. Works for every quadratic.
• Completing the square, useful when you also need the vertex form.

The discriminant b² − 4ac tells you what to expect: positive → two real roots, zero → one repeated root, negative → no real roots (two complex ones).

Systems of equations

Two or more equations that must hold at the same time. For example:

x + y = 5 x − y = 1

• Substitution: solve one equation for a variable, plug it into the other.
• Elimination: add or subtract equations to cancel a variable. Here, adding gives 2x = 6 → x = 3, then y = 2.

A system can have one solution, no solution (parallel lines), or infinitely many (the same line written twice).

Inequalities and differential equations — related but different

An inequality (like 2x + 1 > 7) uses <, >, ≤, ≥ instead of "=". The solving steps are similar, but multiplying or dividing by a negative number flips the inequality sign.

A differential equation contains derivatives of an unknown function rather than just the variable itself. These are a separate class with their own methods and are not solved by simple algebraic rearrangement.

Solving with an online step-by-step tool

MathDF's solver accepts equations, inequalities, systems, derivatives, integrals, limits, and matrices in one input field. According to the site, it works like this:

1. Enter the problem — type it, build it with the math keyboard, photograph or draw it, or describe it in words.
2. Get an instant answer — every change to the input is analyzed in your browser; if it contains a math expression, the built-in calculator solves it immediately under the field, before anything is sent to the AI. Results such as roots and simplified forms appear as you type, and you can tap a result to insert it back into the input.
3. Open the step-by-step solution — if the problem can be solved step by step, a link button lights up and opens the matching calculator with your expression already loaded, with no AI involved.
4. Use photo or handwriting recognition — point the camera at a problem or draw it by hand; recognition converts it to an expression in the input, where you can fix a symbol before solving.
5. Ask the AI chat — text and images go to the assistant for explanations, word problems, proofs, or checking your own solution. There are two assistant modes (a math-oriented one and a general-purpose one); the mode button only changes the system prompt, and you can switch at any time.

The site's own examples show the instant-answer behavior: x² − 5x + 6 = 0 returns x₁ = 2, x₂ = 3, and the system x + y = 5, x − y = 1 returns x = 3, y = 2.

This is most useful when you already have a method in mind and want to verify each transformation, or when you want to see the intermediate steps for a problem type you're learning.

Always verify your solution

Substitution is the check that catches most mistakes:

1. Take each value you found.
2. Put it back into the original equation (not a rearranged version).
3. Confirm both sides are equal.

For the system above: 3 + 2 = 5 ✓ and 3 − 2 = 1 ✓.

Watch for these cases:

• No solution — you reach a contradiction like 0 = 5.
• Infinitely many solutions — you reach an identity like 0 = 0, meaning any value works.
• Extraneous roots — if you squared both sides or cleared a denominator, test every candidate; some may not satisfy the original equation.
• Division by zero — exclude any value that makes a denominator zero, even if it appears in your result.

Quick reference

Pick the method by the highest power of the unknown and the number of equations. Solve by hand for understanding, then use a step-by-step solver to confirm each step and catch sign errors before they compound.