Simultaneous equations and inequalities

01 / UNDERSTAND & APPLY

A solution must satisfy all equations together

A system is a collection of conditions that apply simultaneously. A pair (x,y) solving one equation is not enough; it must satisfy every equation and all domain restrictions. Two linear equations in two variables can have one solution, no solutions, or infinitely many.

Choose variables with units, and keep their meanings unchanged. An equation for total objects and another for total cost are independent conditions; writing the same relationship twice in different forms supplies no additional information.

02 / UNDERSTAND & APPLY

Replace a variable by an equal expression

Substitution is efficient when one equation already isolates a variable or has coefficient 1 or −1. Solve that equation for a variable, then replace every occurrence of that variable in the other equation. Parenthesize the substituted expression so all signs and factors distribute correctly.

Once one value is found, substitute back for the other, and verify the ordered pair in both originals. The order in (x,y) matters.

03 / UNDERSTAND & APPLY

Choose combinations that remove a variable

Adding equations preserves common solutions. Multiply one or both equations by nonzero constants so coefficients of a chosen variable are opposites, then add. Multiply every term, including the right side. Subtraction is equally valid but requires careful handling of the entire second equation.

If the question asks for a sum or difference, an equation combination may produce it directly. Solving for each individual variable can be unnecessary.

04 / UNDERSTAND & APPLY

Parallel, intersecting or identical lines

For nonvertical lines written y=mx+b, different slopes give one intersection. Equal slopes with different intercepts give distinct parallel lines and no solution. Equal slopes and equal intercepts give the same line and infinitely many solutions. Algebraically these cases yield a unique value, a contradiction, or an identity.

A zero coefficient requires care; vertical lines cannot be written with a finite slope. More generally, compare whole equations, including constants. Proportional left-hand coefficients do not guarantee dependent equations if their right-hand constants are not proportional.

05 / UNDERSTAND & APPLY

The requested expression may be fixed even when variables are not

If every solution satisfies x+y=8, the sum is known even though x and y may vary. A dependent system can therefore determine a requested expression without determining the pair. Conversely, knowing a sum does not fix the product or difference.

For QC, distinguish lack of a unique variable from lack of a unique comparison. Use equation combinations to see what remains fixed, then use two valid solutions to test what varies.

06 / UNDERSTAND & APPLY

Intersect the feasible regions

Simultaneous inequalities require their intersection. In one variable, solve each and keep only their common interval. In two variables, each linear inequality describes a half-plane; include its boundary for ≤ or ≥ and exclude it for < or >.

To identify the side of a line, test a point not on the boundary, such as the origin if allowed. A dashed boundary signals exclusion, a solid one inclusion. The overlap of shaded regions contains feasible points; a boundary intersection alone does not establish that a point satisfies all restrictions.

07 / UNDERSTAND & APPLY

Two kinds of totals often supply two equations

A standard model uses x+y=N for the count and px+qy=C for a cost, mass or revenue total. Another uses distance=rate×time for each traveler plus a shared distance or time relationship. Units must match across each equation.

Check feasibility after algebra. A ticket count of −2 or a fractional number of indivisible items cannot describe the stated situation. If a system is inconsistent, do not choose the “closest” pair unless the question explicitly permits approximation.

08 / UNDERSTAND & APPLY

Choose a method and answer the requested question

All examples and practice in this lesson are original. For quantitative comparison (QC), use A if Quantity A is greater, B if Quantity B is greater, C if they are equal, and D if the relationship cannot be determined from the information given. A, B or C must hold for every permitted case. To establish D, give two allowed cases producing different relationships; an inability to calculate is not itself evidence for D.

A single-answer question requires one choice. For multiple-answer questions, obey the stated selection count or select every correct option if instructed. Test each option separately; do not stop after finding one. Multiple-answer questions receive credit only when the complete correct set is selected, with no incorrect choices. Numeric entry requires the requested value, with units handled in the calculation; round only when the question directs you to. A single numeric-entry box accepts an integer or decimal; a pair of fraction boxes accepts an integer numerator and a nonzero integer denominator. Fractions need not be reduced, but reducing makes checks easier. Do not enter a percent sign when the requested number is a percentage. Preserve exact fractions until the last step.

A calculator can check arithmetic but cannot decide the domain, interpretation or denominator for you. Estimate sign and size first, enter parentheses carefully, and compare the result with the original conditions. Cancel factors or exploit structure when doing so is faster and more reliable than decimal entry.

09 / UNDERSTAND & APPLY

Original practice: simultaneous means every condition

Check answers in all original equations.