Number & Algebra
Linear Relations and Equations
Grade 7 · Grade 8 · Grade 9
- ✓By the end of this lesson students will be able to solve linear equations using inverse operations.
- ✓By the end of this lesson students will be able to graph linear relations from tables of values or equations.
- ✓By the end of this lesson students will be able to identify and classify polynomials.
- ✓By the end of this lesson students will be able to add and subtract simple polynomials.
Key concepts
A relationship between two variables that, when graphed on a coordinate plane, forms a straight line. The change in one variable is directly proportional to the change in the other.
A mathematical statement that shows two expressions are equal. It always contains an equals sign (=).
The process of finding the value(s) of the variable(s) that make the equation true. This is typically done by isolating the variable on one side of the equation.
Operations that undo each other. For example, addition is the inverse of subtraction, and multiplication is the inverse of division. These are used to isolate variables when solving equations.
The visual representation of a linear relation on a coordinate plane. Points (x, y) that satisfy the relation are plotted and connected to form a straight line.
An algebraic expression consisting of one or more terms, where each term is a product of a constant and one or more variables raised to non-negative integer powers.
A single number, a variable, or a product of numbers and variables. Terms in a polynomial are separated by addition or subtraction signs.
The numerical factor of a term in a polynomial. For example, in the term 5x², 5 is the coefficient.
A term in a polynomial that does not contain a variable. Its value remains constant.
The sum of the exponents of the variables in a single term. For example, the degree of 3x²y is 2+1=3.
The highest degree of any term in the polynomial. This determines the overall degree of the polynomial.
Polynomials are classified by the number of terms: Monomial (one term), Binomial (two terms), Trinomial (three terms).
Key facts to remember
- 1A linear relation always produces a straight line when graphed.
- 2To solve an equation, use inverse operations to isolate the variable.
- 3The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope and 'b' is the y-intercept.
- 4Polynomials are algebraic expressions where variables have non-negative integer exponents.
- 5Like terms have the exact same variables raised to the exact same powers and can be combined by adding or subtracting their coefficients.
- 6The degree of a polynomial is determined by the highest exponent of any variable in any term.
Worked examples
Example 1
Solve for x: 4x - 9 = 15
Answer
x = 6
Always perform the same operation on both sides of the equation to maintain balance.
Example 2
Graph the linear relation y = -x + 3.
Answer
A straight line passing through the points (-1, 4), (0, 3), and (2, 1). The line has a negative slope and crosses the y-axis at 3.
Using at least three points helps ensure accuracy and confirms the relation is linear.
Example 3
Simplify the polynomial expression: (7x² - 2x + 5) - (3x² + 4x - 1)
Answer
4x² - 6x + 6
Be careful with signs when subtracting polynomials; a negative outside the parentheses changes the sign of every term inside.
Common mistakes
- ✗Forgetting to apply an operation to both sides of an equation, leading to an incorrect solution.
- ✗Incorrectly combining unlike terms (e.g., adding x and x²). Remember, only like terms can be combined.
- ✗Making sign errors, especially when distributing a negative sign during polynomial subtraction.
- ✗Plotting points incorrectly on the coordinate plane, which results in a graph that does not represent the relation.
- ✗Not extending the line with arrows when graphing a linear relation, implying the line stops at the plotted points rather than continuing infinitely.
Exam tips
- ★Always check your solution to an equation by substituting the value back into the original equation to ensure it makes the statement true.
- ★When graphing, use a ruler to draw straight lines and clearly label your axes (x and y) and scale.
- ★Show all your steps clearly when solving equations or simplifying expressions to earn full marks, even if you can do some steps mentally.
- ★Practice identifying like terms before attempting to add or subtract polynomials to avoid errors.
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