Solving linear equations with algebra and graphs

Algebra • Solving equations and inequalities

Flashcards

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What is the gradient of y = −3x + 2 and how does it appear on the graph?

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The gradient is −3 and the line slopes down three units vertically for each one unit to the right.

Key concepts

What you'll likely be quizzed about

Definition of a linear equation in one unknown

A linear equation in one unknown has the form ax + b = 0 or can be rearranged to that form, where a and b are constants and a is not zero. The graph of y = ax + b is a straight line. The solution of ax + b = 0 is the value of x that makes the expression equal to zero. Limiting factors include a = 0, which changes the equation to a constant and requires separate handling.

Inverse operations and balancing

Algebraic solution relies on inverse operations to isolate the unknown. Addition cancels subtraction and multiplication cancels division. Each operation applied to one side of the equation must also apply to the other side to maintain equality. Cause: a non-isolated variable; Effect: apply inverse operations step-by-step until the variable stands alone.

Solving equations with the unknown on both sides

Collect like terms by moving variable terms to one side and constant terms to the other. Cause: variable terms on both sides create ambiguity; Effect: subtract or add the same variable term from both sides, then divide or multiply to find the value. Special cases arise: identical expressions on both sides produce infinitely many solutions; contradictory constants produce no solution.

Checking solutions

Substitution of the found value into the original equation confirms correctness. Cause: algebraic manipulation can produce arithmetic errors or loss of solutions; Effect: direct substitution verifies that both sides of the original equation are equal for the proposed solution.

Linear equations on a graph

Representation of a linear equation as y = mx + c produces a straight line with gradient m and y-intercept c. Cause: a linear relationship between x and y; Effect: plotting points and drawing a straight line provides a visual model. Solving ax + b = 0 corresponds to finding the x-intercept of y = ax + b, since the x-intercept is the point where y = 0.

Approximate solutions from graphs

Graphical methods produce approximate values by reading coordinates at intersections or intercepts. Cause: limited scale and drawing accuracy; Effect: the solution is approximate and depends on the plotting precision. Use finer scales, more plotting points, or algebraic methods to obtain exact values where needed.

Key notes

Important points to keep in mind

Isolate the variable by applying inverse operations to both sides to maintain equality.

Move all variable terms to one side and constants to the other before simplifying.

Check for special cases: identical expressions imply infinitely many solutions; contradictions imply no solution.

Convert linear equations to y = mx + c to use graphical methods; the x-intercept gives solution for ax + b = 0.

Graphical solutions are approximate; use finer scales or algebraic methods for exact answers.

Always substitute the proposed solution into the original equation to verify correctness.

Avoid dividing by zero; treat any step that requires it as an invalid operation and re-examine the equation.

When reading coordinates from a graph, consider the plotting scale and round to the required precision.