Create and simplify linear expressions

Can three x plus five be collected into eight x for every value of x?

Read terms with their signs

In the linear expressions here, a variable term is an integer coefficient multiplied by one letter, and a constant term has no letter

The coefficients are four and negative three; the constant is seven. Keep the minus sign with the term it belongs to

An unprinted coefficient is one, not zero. A minus sign by itself before a variable represents coefficient negative one

A variable represents a number, not a unit label. Different letters may take different values

Term

\(\displaystyle -p\)

Enter the signed coefficient of the variable term

Write the term as an integer multiplied by p

Enter the constant term, including its sign

Find the term with no variable and keep its preceding sign

Which pair can be collected into one term by adding its coefficients?

Reorder additions while retaining signed terms

Commutativity allows addends to change order, and associativity allows them to be regrouped

First interpret subtraction as addition of the opposite term. Then move the entire signed term, not its unsigned magnitude

Subtraction itself is not commutative: five minus two is not two minus five

The corresponding multiplication laws allow factors to be reordered and regrouped. They do not turn a sum into a product

Original expression

\(\displaystyle 6a-4b+2a\)

Which rearrangement preserves the given expression for all a and b?

Zero and one play different identity roles

Zero is the additive identity: adding it leaves the number unchanged

One is the multiplicative identity: multiplying by it leaves the number unchanged

Multiplication by zero is not an identity operation. It makes the product zero

These facts explain why coefficient one may be omitted and why a zero-coefficient group disappears from a simplified expression

An inverse combines to give the relevant identity

The additive inverse of a is its opposite, negative a. Their sum is the additive identity zero

The multiplicative inverse of a non-zero number is its reciprocal. Their product is the multiplicative identity one

Zero has an additive inverse, itself, but no multiplicative inverse because zero times any number is zero, not one

Given value

\(\displaystyle -7\)

Which number added to the given value makes zero?

Use the opposite signed value

Given value

\(\displaystyle -4\)

Which exact fraction multiplied by the given value makes one?

Use the reciprocal, retaining the sign needed for a positive product

Use distributivity to justify collecting

Both terms are multiples of the same variable. Distributivity lets their numerical coefficients be added while the variable stays the same

Use signed addition for subtraction as well. A negative coefficient is a valid result

The answer tables ask for signed integer coefficients. Enter zero for a requested group that cancels; do not leave its box empty

Collect two positive multiples of one variable

Collect the like terms

Enter signed integers in this order: coefficient of x. Enter 0 for a group that cancels completely

Start with the given expression

Add the signed coefficients of this pronumeral group

Construct a combined coefficient

Allow a negative combined coefficient

Collect the like terms

Enter signed integers in this order: coefficient of x. Enter 0 for a group that cancels completely

Start with the given expression

Add the signed coefficients of this pronumeral group

Add coefficients with different signs

Include the coefficient hidden in a minus sign

Recognise additive inverses in a variable group

Collect the like terms

Enter signed integers in this order: coefficient of x. Enter 0 for a group that cancels completely

Start with the given expression

Add the signed coefficients of this pronumeral group. Its zero coefficient removes the group

Collect constants separately from variable terms

The x coefficients combine with each other. The constants combine with each other

Do not attach an x to the constant total or add that total to the x coefficient

A variable group can cancel while a non-zero constant remains

Construct both the variable coefficient and constant

Collect the like terms

Enter signed integers in this order: coefficient of x, constant term. Enter 0 for a group that cancels completely

Start with the given expression

Write matching factors in the same order and bring like terms together without changing their signs

Add the signed coefficients of this pronumeral group

Add the signed constants

Retain negative results in both groups

Enter zero for a cancelled variable group

Keep unlike variable groups separate

The x coefficients add within the x group, and the y coefficients add within the y group

The final two terms are unlike. Their coefficients cannot be added to make one term such as 8xy

An absent printed coefficient before x means one. When a group cancels, its zero coefficient still belongs in the requested answer table

Construct separate coefficients for two variables

Collect the like terms

Enter signed integers in this order: coefficient of x, coefficient of y. Enter 0 for a group that cancels completely

Start with the given expression

Write matching factors in the same order and bring like terms together without changing their signs

Add the signed coefficients of this pronumeral group

Add the signed coefficients of this pronumeral group

Combine two signed groups

Retain two variable totals and a constant

Do not drop the uncancelled group

Construct both zero coefficients when all groups cancel

Use substitution as a check, not a universal proof

Equivalent expressions give the same value for every allowed choice of their variables. The arithmetic laws justify the collection steps for all those choices

Substituting a convenient pair of values can detect an error. One mismatch disproves equivalence, but matching at a few chosen values alone does not prove an identity

Keep algebraic justification and numerical checking as separate parts of your explanation

Expression

\(\displaystyle 2x+6y\)

Values

\(\displaystyle x=2,\quad y=-1\)

Evaluate the collected expression at the supplied values

Multiply each coefficient by its variable value before adding

Original

\(\displaystyle 3x+2y-x+4y\)

Collected

\(\displaystyle 2x+6y\)

Change x and y and compare both expression values. The difference is original minus collected. All slider values are integers

What justifies equivalence for every x and y, beyond the sampled table values?

Translate quantities before collecting their terms

If one notebook costs x dollars, three identical notebooks cost 3x dollars. Two more at the same unit price add 2x dollars

State the same-unit-price assumption. If a different item costs y dollars, keep its y term separate from the x terms

A fixed delivery fee is a constant term. For example, five notebooks at x dollars each and a four-dollar delivery fee cost 5x + 4 dollars

Choose the variable meanings and original expression before simplifying; collecting cannot repair an incorrect translation

Interpret the combined variable coefficient in a cost expression

Create, simplify and justify your own expression

Independent written task, not automatically marked

Invent two item types with unit prices represented by different letters and one fixed fee. State what each letter and each unit means

Describe two orders containing those items, write an expression for their combined cost and simplify it by grouping like terms

Explain every collection step using the signed coefficients and arithmetic laws. State whether a term is a variable term or a constant

Choose positive sample prices to check your contextual expression numerically, then explain why your algebraic simplification does not depend on those particular prices

Separately construct a signed linear expression with a cancelling variable group. Simplify it, retain any uncancelled groups and explain which inverse property is used

Review the formulation and explanation against these instructions or with a teacher; the coefficient tables check supplied expressions rather than your independent creation

Preserve the terms while simplifying their form

Read every term with its sign. A hidden coefficient is one, and a minus sign before a letter means coefficient negative one

Move signed addends using commutativity and associativity; use distributivity to combine matching variable terms

Collect constants separately. Zero coefficients represent cancelled groups, while unlike variables remain separate

Adding zero and multiplying by one preserve a number. Opposites add to zero, and non-zero reciprocals multiply to one

Use substitution to check, and arithmetic laws to justify an identity for all allowed values

When creating an expression, define its quantities and assumptions before collecting its terms