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MAT 105 · Unit 1 ALEKS · objective 1.2

What an Equation and Solution Mean

Interpret an equation as a true-or-false balance statement and identify values that satisfy it.

Standalone concept package
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Start here

Build the concept from the beginning

Starting point

No prior equation-solving skill is assumed. Arithmetic operations, variables, equality, and every new algebra decision are explained on this page.

Learning target

Interpret an equation as a true-or-false balance statement and identify values that satisfy it.

Evidence of mastery

Construct the reasoning path, solve at least 4 of 5 mastery problems, and revise an unfamiliar explanation.

Essential terms

equation
A statement that two mathematical expressions have the same value.
solution
A value that makes an equation true when substituted for the variable.
coefficient
A numerical factor multiplying a variable.
constant
A number without a variable.
Look first, then name the mathematics

Build the visual meaning

A level balance holds the expression 3x + 2 on one side and 14 on the other, with x = 4 shown as the value that makes both sides equal.
How to read this visual: The balance visual pairs equal expressions and shows that a solution is the value that keeps the balance level.
  • Left side
  • Right side
  • x = 4
  • Both sides = 14
  • Solution
Read the complete visual relationship as text
  • Left expression: 3x + 2
  • Right expression: 14
  • Test x = 4
  • Both sides = 14
  • True equation
Definition in plain language

An equation says that its left expression and right expression have equal value. Solving means finding every allowed value that makes that statement true.

Why this matters

Every later solving method preserves this central idea: the two complete sides must remain equal.

Reasoning path

See the method as a sequence

Move from left to right. Each decision prepares the next one; on a phone, follow the numbered cards from top to bottom.

Locate the equal sign and separate the two expressions.
Identify the unknown variable and the operations acting on it.
Test a proposed value by substitution.
Decide whether the resulting numerical statement is true.

The same method in words

  1. Locate the equal sign and separate the two expressions.
  2. Identify the unknown variable and the operations acting on it.
  3. Test a proposed value by substitution.
  4. Decide whether the resulting numerical statement is true.
Interactive decision model

Build the reasoning, one decision at a time

Build a valid solution check for 2x + 3 = 13.

Choose this step, then check the construction.
Choose this step, then check the construction.
Choose this step, then check the construction.

The reasoning path changes as each decision is selected.

Model before practice

Three fully explained examples

Every example identifies the goal, explains the next move, carries out the algebra, and checks or interprets the result.

Test x = 5 in 2x + 1 = 11

Name the goalThe goal is not to isolate x; it is to decide whether 5 is a solution.
Choose the next moveReplace every x with 5 and keep the rest of the equation unchanged.
Carry out the mathematics2(5) + 1 = 10 + 1 = 11, so the equation becomes 11 = 11.
Check and interpretThe final statement is true, so x = 5 satisfies the equation.

Test x = −2 in x² = 4

Name the goalA negative value can still be a solution when the variable is squared.
Choose the next moveSubstitute −2 using parentheses so the sign is included in the square.
Carry out the mathematics(−2)² = 4, so the equation becomes 4 = 4.
Check and interpretThe statement is true. Both x = −2 and x = 2 solve x² = 4.

Interpret 4y − 3 = 17

Name the goalIdentify the variable, coefficient, constant, and the equality claim.
Choose the next moveRead 4y as four times an unknown value, then subtract 3.
Carry out the mathematicsThe equation claims that the entire left expression has the same value as 17.
Check and interpretA solution is any y that makes 4y − 3 evaluate to 17; y = 5 works.

Practice with decreasing support

Guided practice

Practice immediately after the model

Predict first. Then use the visual relationship and reasoned feedback to check two decisions.

1. In 3x + 5 = 17, what does the equal sign state?

2. Which value makes x + 4 = 11 true?

Not completed yet.
Partially completed problem

Finish the missing step

A prompt supplies part of the reasoning. Predict the missing mathematical move before choices appear.

1. What is a solution of an equation?

Not completed yet.
Independent practice

Six problems for repetition

Solve all six. Use the specific feedback to revise a method or sign error.

1. Which statement is an equation?

2. In 5y − 2, which symbol is the variable?

3. In 7x + 3, what is the coefficient of x?

4. In 4x + 9 = 21, which expression is the left side?

5. If x = 6, what does x − 1 evaluate to?

6. Which value does not satisfy x² = 9?

Not completed yet.

Mastery and transfer

Mastery check

Demonstrate understanding

Attempt all five. At least four correct answers are required for mastery.

1. Why can an equation have more than one solution?

2. What should be checked after solving an equation?

3. Which equation is true when x = 4?

4. What does it mean if no value makes an equation true?

5. What does it mean if every allowed value makes an equation true?

Not completed yet.
Unfamiliar transfer

Explain, compare, and revise

Explain why x = 3 is a solution of 4x − 2 = 10 but x = 2 is not.

Write a first attempt before opening the self-check.

The record reveals exact answers and model reasoning only after the corresponding attempt or self-check has occurred.