2.1 Algebra & Equations

Introduction to Algebra

Algebra is the foundational language of higher mathematics. It allows us to represent unknown values with variables and solve for them using established rules. A strong grasp of algebra is essential for tackling more advanced mathematical concepts on the DCAT, as it provides the core toolkit for manipulating and evaluating equations. By mastering algebra, you learn how to structurally approach a problem and find the missing piece of the puzzle.

Linear Equations

A linear equation is an equation of the first degree, meaning the highest power of the variable is 1. The standard form is $ax + b = c$, where $a, b,$ and $c$ are real numbers and $a \neq 0$.

Solving Single-Variable Linear Equations

To solve a linear equation, the goal is to isolate the variable on one side of the equation. You do this by performing inverse operations. The golden rule of algebra is: Whatever you do to one side of the equation, you must do to the other to maintain equality. If you add to the left side, you add to the right side.

Example: Solve for $x$ in the equation $3x - 7 = 14$.

1. First, isolate the term with the variable. Add 7 to both sides: $3x = 14 + 7 \implies 3x = 21$
2. Next, isolate $x$ by dividing by the coefficient. Divide by 3: $x = 21 / 3 \implies x = 7$

Linear Equations with Variables on Both Sides

When a variable appears on both sides of the equal sign, first group all variable terms on one side and all constant terms on the other. This usually involves adding or subtracting terms across the equal sign.

Example: Solve for $x$ in $5x + 2 = 2x + 11$.

1. Subtract $2x$ from both sides to group the $x$ terms on the left: $3x + 2 = 11$
2. Subtract 2 from both sides to group the constants on the right: $3x = 9$
3. Divide by 3 to isolate $x$: $x = 3$

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Quadratic Equations

A quadratic equation is a second-degree polynomial equation. Its standard form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \neq 0$. The graph of a quadratic equation forms a parabola, and solving the equation typically means finding the x-intercepts (the roots).

Solving by Factoring

Factoring involves expressing the quadratic polynomial as a product of two binomials. This relies on the zero-product property: if $a \times b = 0$, then $a = 0$ or $b = 0$.

Example: Solve $x^2 - 5x + 6 = 0$.

1. Find two numbers that multiply to the constant term (6) and add to the coefficient of the middle term (-5). These numbers are -2 and -3.
2. Write the quadratic as a product of binomials: $(x - 2)(x - 3) = 0$
3. Set each factor to zero to find the solutions: $x - 2 = 0 \implies x = 2$ and $x - 3 = 0 \implies x = 3$

Special Factoring Patterns

Recognizing patterns can save time. The difference of squares is $a^2 - b^2 = (a - b)(a + b)$. A perfect square trinomial is $a^2 + 2ab + b^2 = (a + b)^2$.

The Quadratic Formula

When a quadratic equation cannot be easily factored, you can always use the quadratic formula to find the roots:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The term under the square root, $b^2 - 4ac$, is called the discriminant. It tells you the nature of the roots:

• If $b^2 - 4ac > 0$, there are two distinct real roots.
• If $b^2 - 4ac = 0$, there is exactly one real root.
• If $b^2 - 4ac < 0$, there are two complex roots (no real roots).

Example: Solve $x^2 + 4x - 5 = 0$.

Here, $a = 1$, $b = 4$, and $c = -5$.

1. Calculate the discriminant: $4^2 - 4(1)(-5) = 16 + 20 = 36$.
2. Apply the formula: $x = \frac{-4 \pm \sqrt{36}}{2(1)} = \frac{-4 \pm 6}{2}$.
3. This gives two solutions: $x = \frac{2}{2} = 1$ and $x = \frac{-10}{2} = -5$.

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Systems of Linear Equations

A system of linear equations consists of two or more equations with the same set of variables. The solution to the system is the set of values that satisfies all equations simultaneously. Graphically, this is the point where the lines intersect.

Substitution Method

This method is best when one variable is already isolated or has a coefficient of 1.

1. Isolate one variable in one of the equations.
2. Substitute the resulting expression into the other equation.
3. Solve for the remaining variable.
4. Substitute this value back into the isolated equation to find the other variable.

Example: Solve the system: $x + y = 6$ $2x - y = 3$

From the first equation, we can express x as $x = 6 - y$. Substitute this into the second equation: $2(6 - y) - y = 3 \implies 12 - 2y - y = 3 \implies 12 - 3y = 3 \implies 3y = 9 \implies y = 3$. Now, substitute y back: $x = 6 - 3 = 3$. The solution is $(3, 3)$.

Elimination Method

This method is best when both equations are in standard form.

1. Multiply one or both equations by a constant so that the coefficients of one variable are opposites (e.g., 2x and -2x).
2. Add the equations together to eliminate that variable.
3. Solve for the remaining variable.
4. Substitute back into an original equation to find the other variable.

Example: Solve the system: $3x + 2y = 16$ $x - 2y = 0$

Notice the y coefficients are opposites. Add the equations directly: $(3x + x) + (2y - 2y) = 16 + 0 \implies 4x = 16 \implies x = 4$. Substitute $x=4$ into the second equation: $4 - 2y = 0 \implies 2y = 4 \implies y = 2$. The solution is $(4, 2)$.

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Inequalities

Solving linear inequalities is very similar to solving linear equations, with one critical exception that must be memorized: When you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

Example: Solve $-2x + 5 < 11$.

1. Subtract 5 from both sides: $-2x < 6$
2. Divide by -2. Since we are dividing by a negative number, flip the sign: $x > -3$

Compound Inequalities

A compound inequality consists of two inequalities joined by "and" or "or". For an "and" inequality like $-4 < 2x \le 10$, you solve all parts simultaneously by dividing all three sections by 2 to get $-2 < x \le 5$.

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Exponents and Radicals

Understanding the rules of exponents and radicals is necessary for simplifying algebraic expressions efficiently.

Rules of Exponents

• Product Rule: $x^a \cdot x^b = x^{a+b}$ (When multiplying like bases, add the exponents).
• Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$ (When dividing like bases, subtract the exponents).
• Power of a Power: $(x^a)^b = x^{ab}$ (When raising a power to another power, multiply them).
• Negative Exponent: $x^{-a} = \frac{1}{x^a}$ (A negative exponent indicates a reciprocal).
• Zero Exponent: $x^0 = 1$ (Any non-zero base raised to the power of zero is 1).

Radicals and Fractional Exponents

Radicals are the inverse operation of exponents. A fractional exponent represents a radical: $x^{\frac{1}{n}} = \sqrt[n]{x}$, and more generally $x^{\frac{m}{n}} = \sqrt[n]{x^m}$.

To simplify a square root, factor out perfect squares. Example: Simplify $\sqrt{48}$. $\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}$.

Common Traps to Avoid

• Trap: Assuming $(a+b)^2 = a^2 + b^2$. This is mathematically incorrect. You must foil the binomial. The correct expansion is $(a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2$.
• Trap: Forgetting to flip the inequality sign when dividing by a negative number. This is one of the most common mistakes in algebra.