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Welcome to GCSE Edexcel Maths revision.

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Topic M 10: Linear equations and inequalities.

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This video covers Higher tier.

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It includes the shared content and the labelled Higher extensions.

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An equation's solution makes both sides equal.

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Keep an equation balanced by doing the same operation to each side; inverse operations help isolate the unknown.

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Worked example: 5 x minus 7 equals 18 gives 5 x equals 25 and x equals 5.

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Check: 5 multiplied by 5 minus 7 equals 18.

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Worked example: 4 multiplied by open bracket x minus 2 close bracket equals 2 x plus 10 gives 4 x minus 8 equals 2 x plus 10, then 2 x equals 18, so x equals 9.

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Expand and collect before dividing.

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For equations with fractions,

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multiply every term by a common denominator: the fraction with numerator open bracket x close bracket and denominator open bracket 3 close bracket ,

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end fraction plus 2 equals 5 gives x plus 6 equals 15,

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so x equals 9.

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To form an equation, define the unknown.

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Three consecutive integers can be n, n plus 1 and n plus 2; if their sum is 24, 3 n plus 3 equals 24, so they are 7, 8 and 9.

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Graphically, solve f of open bracket x close bracket equals g of open bracket x close bracket by reading the x-coordinates of intersections.

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The graph gives approximate solutions if its scale does not allow exact readings.

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Solve a linear inequality like an equation, except multiplying or dividing by a negative number reverses the inequality sign.

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For minus 2 x is less than 6, x is greater than minus 3.

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Worked example: 3 x plus 2 is less than or equal to 14 gives 3 x is less than or equal to 12 and x is less than or equal to 4.

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Equality is allowed, so 4 is included.

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On a number line, use an open circle for is less than or is greater than and a filled circle for is less than or equal to or is greater than or equal to .

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Shade or draw an arrow in the direction of the allowed values.

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Number line for x less than or equal to four

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For a double inequality, apply each operation to all three parts: 2 is less than 3 x plus 5 is less than or equal to 11 gives minus 1 is less than x is less than or equal to 2.

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The integer solutions are 0, 1 and 2.

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The answer x is less than 4 describes infinitely many real numbers unless the question restricts x to integers.

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List integer solutions only when asked.

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Set notation {x to x is greater than or equal to 2} means the set of x values satisfying x is greater than or equal to 2.

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Intersection means all conditions apply at once; union combines values allowed by either condition.

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For an inequality in two variables, first draw its boundary line.

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Use a solid line if equality is allowed and a dashed line if it is not.

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Test a point off the line to decide which side is allowed.

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For y is greater than 2 x plus 1, open bracket 0, 0 close bracket fails, so the region is on the other side of the boundary.

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For simultaneous inequalities, the solution is their overlapping region.

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State clearly whether the shaded region is the allowed region or the excluded region.

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That completes Linear equations and inequalities.

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Revisit the notes and test yourself on the revision website.