Algebra 1 course

1. 1.Linear Equations

   A linear equation in one variable has the variable only to the first power, such as $3(x - 2) + 5 = 2x + 7$. Solving it means simplifying each side, collecting the variable terms on one side and the constants on the other, then dividing by the coefficient.

   • Distribute and combine like terms on each side before moving anything.
   • Move variable terms to the side where the coefficient stays positive to reduce sign errors.
   • Clear fractions by multiplying every term by the least common denominator.
   Quick check: Solve $5x + 3 = 2x - 9$.

   $x = -4$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study linear equations

2. 2.Slope and Lines

   Slope measures how steep a line is: the change in $y$ divided by the change in $x$ between any two points, $m = \frac{y_2 - y_1}{x_2 - x_1}$. Because a line has constant steepness, every pair of points gives the same slope, which is what makes a straight line straight.

   • Positive slope rises left to right, negative slope falls, zero slope is horizontal, and a vertical line has undefined slope.
   • Parallel lines have equal slopes.
   • Perpendicular lines have slopes whose product is $-1$, such as $2$ and $-\frac{1}{2}$.
   Quick check: Find the slope through $(1, 4)$ and $(5, 12)$.

   $2$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study slope and lines

3. 3.Systems of Equations

   A system of two linear equations asks for the point that satisfies both at once, which is where their lines cross. Substitution solves one equation for a variable and plugs it into the other; elimination adds or subtracts multiples of the equations so one variable disappears.

   • Choose substitution when a variable already has coefficient 1 or $-1$.
   • Choose elimination when coefficients line up or can be matched with small multipliers.
   • Always check the pair in both original equations.
   Quick check: Solve $x + y = 10$, $x - y = 4$. What is $x$?

   $x = 7$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study systems of equations

4. 4.Polynomials and Factoring

   A polynomial is a sum of terms of the form $a x^n$ with whole-number exponents, such as $4x^3 - 2x + 7$. The degree is the highest exponent, and it controls the shape of the graph: the end behaviour, the maximum number of turning points and the maximum number of real roots.

   • Always factor out the greatest common factor first.
   • Recognise special products: $a^2 - b^2 = (a - b)(a + b)$ and $a^2 \pm 2ab + b^2 = (a \pm b)^2$.
   • The remainder theorem: dividing $p(x)$ by $x - r$ leaves remainder $p(r)$.
   Quick check: What is the coefficient of $x$ in $(x + 4)(x - 7)$?

   $-3$

   Includes a worked example, 3 common mistakes and 3 practice questions with answers. Study polynomials and factoring

These topics belong to the course outline. Full explainers for them have not been written yet, so they are summarised here rather than given pages of their own.

Linear Inequalities

Solve and graph inequalities; flip the sign when multiplying or dividing by a negative.

• Algebra 1 Practice Test · Ten questions on equations, inequalities, lines, systems, exponents and factoring.

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