Algebra - Solving equations and inequalities: solve linear equations in one unknown algebraically (including those with the unknown on both sides of the equation); find approximate solutions using a graph

The idea

In a linear equation the unknown appears only to the first power, never squared and never in a denominator. Every step keeps the two sides balanced: whatever you do to one side you do to the other. Gather the unknown on one side, gather the numbers on the other, then undo the multiplications and additions in reverse order. To solve 5x − 4 = 3x + 10, take 3x from both sides to get 2x − 4 = 10, add 4 to both sides to get 2x = 14, then divide by 2 so x = 7. Substituting back gives 31 on each side, which is the check worth doing every time. A graphical version asks you to draw y = 5x − 4 and y = 3x + 10 on one grid and read the x coordinate where the lines cross. That reading is approximate, so quote it to the accuracy the grid supports.

What earns the marks

• A correct first step that does the same thing to both sides, such as subtracting the smaller x term from each side.
• Collecting the unknown on one side and the numbers on the other.
• A correct final division even if an earlier arithmetic slip made the number wrong.
• The final value of the unknown, written as a fraction or decimal when it is not a whole number.
• On a graphical question, draw each straight line accurately and read the x coordinate of their intersection.

Common mistakes and how to put them right

• Moving a term across the equals sign without changing its sign, turning x + 6 = 10 into x = 10 + 6.

• Dividing only one side by the coefficient, so 2x = 14 is written as x = 14.

• Expanding a bracket onto the first term only, so 3(x + 4) becomes 3x + 4.

• Reading the y coordinate of the crossing point when the equation was solved for x.

Exam tips

• Substitute your answer into the original equation before moving on; if the two sides disagree you still have time to find the slip.
• When the equation contains a fraction, multiply every term by the denominator first so you are working with whole numbers.
• On a graph question read to the nearest small square and say that the answer is approximate, because that is what is being asked for.