6 — Systems and Algebraic Equations

Solutions and Validity

An equation states a condition that an unknown quantity must satisfy. A solution set contains every value that makes the original equation true. In a system of equations, the same variables must satisfy all equations simultaneously.

A reliable solving process preserves equivalence whenever possible:

• Add or subtract the same quantity from both sides.

• Multiply or divide both sides by the same nonzero quantity.

• Factor and apply the zero-product property.

• Substitute equivalent expressions.

Before manipulating an equation, identify its domain restrictions. A denominator cannot be zero, a real logarithm requires a positive argument, and an even root requires a nonnegative radicand when working with real numbers. For example, $\frac{x+1}{x-3}=2$ requires $x\ne3$, while $\log_2(x-4)=3$ requires $x>4$.

Some operations can create an extraneous solution, so substitute every proposed answer into the original equation or into every original equation in the system.

Takeaway: Solve while preserving the original conditions, then verify every result in the original problem.

Polynomial Equations

A polynomial equation can be written in the form

When a polynomial factors, use the zero-product property:

For example,

Therefore, the solutions are $x=-1$, $x=1$, and $x=4$.

For a quadratic equation $ax^2+bx+c=0$, choose among factoring, completing the square, and the quadratic formula:

The discriminant $\Delta=b^2-4ac$ predicts the real-solution pattern:

• If $\Delta>0$, there are two distinct real solutions.

• If $\Delta=0$, there is one repeated real solution.

• If $\Delta<0$, there are no real solutions.

For higher-degree polynomials that do not factor easily, the Rational Root Theorem gives possible rational zeros to test. Graphing and numerical methods can approximate remaining real zeros. A degree-$n$ polynomial has at most $n$ real zeros and exactly $n$ complex zeros when multiplicity is counted.

Takeaway: Factor first when possible; otherwise use the quadratic formula, rational-zero tests, graphing, or numerical approximation according to the polynomial's structure.

Rational Equations

Rational equations contain ratios of polynomials, such as

Use this sequence:

1. Set every denominator equal to zero to identify excluded values.

2. Find a common denominator.

3. Multiply through by the least common denominator.

4. Solve the resulting equation.

5. Reject restricted or extraneous values.

6. Check all remaining values in the original equation.

Consider

The restrictions are $x\ne0$ and $x\ne-1$. Multiplying by $x(x+1)$ gives

which simplifies to

The quadratic formula gives $x=1\pm\sqrt{3}$. Neither value violates the restrictions, so the solution set is

A rational equation can have no solution, one solution, or several solutions. Its graph may also have vertical asymptotes, which correspond to excluded domain values.

Takeaway: Clearing denominators simplifies the equation, but it never removes the need to preserve restrictions and check the original equation.

Exponential and Logarithmic Equations

Exponential and logarithmic equations are inverse problems. An exponential function has the form $f(x)=b^x$, where $b>0$ and $b\ne1$. Because it is one-to-one, equal powers with the same valid base have equal exponents.

When bases match, rewrite and compare exponents. For example,

leads to $2x-1=3$, so $x=2$.

When bases do not match conveniently, isolate the exponential expression and use logarithms. For

natural logarithms give

For an equation of the form $Ab^{kx}=C$, first isolate the exponential term:

Then apply a logarithm:

provided the quantities satisfy the relevant domain and sign conditions.

A logarithm is defined by the equivalence

with $b>0$, $b\ne1$, and $x>0$. The product, quotient, and power rules are

For

first require $x-1>0$ and $x+1>0$, so $x>1$. Combining the logarithms and converting to exponential form gives $x^2-1=8$, hence $x=\pm3$. The domain rejects $-3$, leaving $x=3$.

Takeaway: Match exponential bases when possible; otherwise use logarithms. For logarithmic equations, state positive-argument conditions before combining or solving.

Systems and Their Intersections

A system asks for values that satisfy multiple equations simultaneously. If the system is written as

then solve $f(x)=g(x)$, and use the resulting values to recover the corresponding coordinates. Graphically, these solutions are intersection points.

Substitution is effective when one equation already isolates a variable. For

set the right sides equal:

Factoring gives $(x-3)(x+1)=0$, so $x=3$ or $x=-1$. Substitution into $y=2x-1$ produces the solutions $(3,5)$ and $(-1,-3)$.

Elimination is useful when adding or subtracting equations removes a variable. For

adding the equations gives $5x=15$, so $x=3$. Substitution then gives $y=1$, and the solution is $(3,1)$.

For nonlinear systems, several intersections may occur. The system

leads to $x^2=x+2$, with $x=2$ and $x=-1$. Its intersection points are $(2,4)$ and $(-1,1)$.

Takeaway: Choose substitution when a variable is isolated, elimination when terms cancel naturally, and intersection methods when the system is complicated or nonlinear.

Graphical and Numerical Methods

Graphical and numerical methods are valuable when an exact algebraic solution is difficult or when you need to understand the number and location of solutions.

To solve $f(x)=g(x)$ graphically:

1. Rewrite the equations in graphable form.

2. Choose a viewing window that shows the relevant behavior.

3. Locate the intersections or the zeros.

4. Record approximate coordinates.

5. Substitute the approximations into the original equations.

6. State clearly whether the results are exact or approximate.

Equivalently, define $h(x)=f(x)-g(x)$. Then solutions of $f(x)=g(x)$ are the zeros, or $x$-intercepts, of $h(x)$. Graphing can miss an intersection if the window or scale is unsuitable, and it usually supplies decimal approximations rather than exact forms.

The bisection method starts with a continuous function and an interval $[a,b]$ satisfying $h(a)h(b)<0$. It repeatedly calculates $m=\frac{a+b}{2}$, evaluates $h(m)$, and retains the half-interval containing the sign change. For $h(x)=x^3-x-1$, the values $h(1)=-1$ and $h(2)=5$ bracket a root, and repeated bisection gives $x\approx1.325$.

Newton's method uses

It often converges quickly from a good initial estimate, but a poor estimate can cause failure or convergence to an unintended root. A graph or sign-change interval helps choose a starting value.

A reliable numerical workflow is to define the original equation, retain domain restrictions, identify likely solution intervals, run the solver, substitute the result back into the original problem, and report suitable precision.

Takeaway: Use graphs to see behavior and locate candidates, numerical methods to refine approximations, and substitution to verify the reported values.

Method Selection and Modeling

A model translates a situation into variables, functions, equations, and constraints. Apply this process:

1. Define each variable and its units.

2. Identify the appropriate function type.

3. Write equations from the stated relationships.

4. State realistic domain restrictions.

5. Solve algebraically, graphically, or numerically.

6. Interpret the result in context.

7. Check whether the result is reasonable.

A polynomial can model a projectile's height, a rational function can represent a rate involving changing quantities, an exponential function can describe growth or decay, and a logarithmic equation can determine the time required to reach a target amount. When two models describe the same quantity, an intersection identifies when the models agree.

Choose methods strategically:

• Use the zero-product property for a factored polynomial.

• Use factoring, completing the square, or the quadratic formula for a quadratic.

• Use rational-zero tests, graphing, or numerical methods for a difficult higher-degree polynomial.

• State restrictions and clear denominators in a rational equation.

• Match bases for exponential equations when possible.

• Isolate the exponential term and use logarithms when bases differ.

• State the domain, solve, and check for logarithmic equations.

• Use substitution or elimination for suitable systems.

• Use graphical or numerical intersections for complicated nonlinear systems.

Final takeaway: Strong equation solving combines symbolic reasoning, domain awareness, graphical interpretation, numerical accuracy, verification, and sensible interpretation of the result.