Solving Equations: Linear, Quadratic and Simultaneous

You can mark your own work

Here is something true of this subject and of almost no other, and most students never use it. ⚠️ IN MATHS YOU CAN FIND OUT WHETHER YOUR ANSWER IS RIGHT BEFORE ANYBODY MARKS IT. Not guess. Not feel fairly confident. Know. Solving an equation means finding the value that makes it true. So once you have a value, put it back into the equation you started with and work out both sides. ⭐ IF THE TWO SIDES COME OUT EQUAL, YOUR ANSWER IS RIGHT. If they do not, you have just caught a mistake while there is still time to do something about it. That is the whole idea, and it takes about ten seconds. Students skip it for a reason worth naming. ⚠️ CHECKING FEELS LIKE SOMETHING YOU DO AT THE END IF THERE IS TIME LEFT OVER, RATHER THAN PART OF SOLVING. So it gets dropped first when the clock is tight, which is exactly when mistakes are most likely. Three things make this worth building into the way you work, rather than treating as advice. First, it works on everything in this module. ⭐ LINEAR, QUADRATIC AND SIMULTANEOUS EQUATIONS ARE SOLVED IN THREE QUITE DIFFERENT WAYS, BUT THEY ARE ALL CHECKED THE SAME WAY. One habit, three topics. Second, it catches the specific mistakes this topic produces. Sign errors when rearranging, and reading the roots straight off a pair of brackets without flipping the signs, are the two commonest ways to lose marks here, and substituting back exposes both instantly. Third, it tells you where to spend whatever time you have left. ⚠️ A QUESTION YOU HAVE CHECKED IS FINISHED. A QUESTION YOU HAVE NOT IS NOT, and knowing which is which is worth more near the end of a paper than another rushed attempt at something new.

Three shapes, three routes, one check

Read the first three rows to see how differently the three are solved, then read the last row and notice it does not change.

Tap the answers that survive a check

Each line gives an equation and a proposed solution. Tap the TWO where putting the value back in actually works.

• In 3x + 4 = 19, the answer x = 5, because 15 plus 4 is 19
• In x squared minus 5x plus 6 = 0, the root x = 2, because 4 minus 10 plus 6 is 0
• In 2x - 1 = 9, the answer x = 4, because 8 minus 1 is 9
• In x squared minus 4 = 0, the root x = 4, because 16 minus 4 is 0

The sign that flips

A student factorises a quadratic to (x - 2)(x - 3) = 0 and writes down the roots as x = -2 and x = -3. How would substituting back reveal the mistake fastest?

• Putting x = -2 into the original equation gives a value nowhere near zero, so the answer fails its own test straight away
• It would not reveal anything, because both signs are negative and the brackets are negative too
• By expanding the brackets again to see whether they multiply back correctly
• By solving it again with the quadratic formula and comparing

Five words used precisely

Five terms, each defined by what it is. No equation is solved here.

Match each equation to the value that solves it

One word, two jobs, and where the marks are

⚠️ "SUBSTITUTION" MEANS TWO DIFFERENT THINGS IN THIS TOPIC, AND NOBODY WARNS YOU. It is the name of a METHOD for simultaneous equations, where you rearrange one equation and put it into the other to get rid of an unknown. And it is the name of the CHECK, where you put a finished answer back into the equation it came from. ⚠️ SAME WORD, COMPLETELY DIFFERENT MOMENTS. If a question says "solve by substitution" it means the method; the check is something you do afterwards whichever method you used. ⭐ NOW THE THING THAT MAKES CHECKING WORTH REAL MARKS RATHER THAN JUST REASSURANCE. Marks on these questions are split between method and answer. ⚠️ YOUR WORKING EARNS ITS MARKS WHETHER OR NOT THE FINAL NUMBER IS RIGHT, PROVIDED IT IS THERE TO BE SEEN. So never rub out working that led to an answer you have decided against: cross it through with one line instead, and it can still be credited. And because method marks are safe either way, the check tells you something useful about the clock. ⚠️ A QUESTION YOU HAVE SUBSTITUTED BACK INTO IS FINISHED AND WILL NOT REPAY MORE TIME. One you have not checked might still be wrong, so that is where the last few minutes should go. Two habits that make the check quick enough to actually do. ⚠️ ALWAYS CHECK IN THE ORIGINAL EQUATION, NOT IN A REARRANGED LINE OF YOUR OWN. If the mistake happened during rearranging, checking against your own rearrangement will confirm the mistake rather than catch it. ⚠️ AND FOR A PAIR OF SIMULTANEOUS EQUATIONS, CHECK IN BOTH. A pair that fits one equation and not the other is precisely what a slip produces, so testing only one is barely a check at all.

Two solutions, both checked

Start with a linear one. 5x - 3 = 2x + 12 Take 2x from both sides: 3x - 3 = 12. Add 3 to both sides: 3x = 15. Divide both sides by 3: x = 5. ⚠️ NOW DO THE PART ALMOST EVERYBODY SKIPS, AND DO IT IN THE EQUATION YOU WERE GIVEN RATHER THAN IN ONE OF YOUR OWN LINES. Left-hand side: five fives is 25, minus 3, which is 22. Right-hand side: two fives is 10, plus 12, which is 22. ⭐ TWENTY-TWO AND TWENTY-TWO. THE ANSWER IS RIGHT, AND YOU KNOW IT IS RIGHT WITHOUT ANYBODY TELLING YOU. Now a quadratic, where there is more to check. x^2 - 7x + 12 = 0 Two numbers that multiply to 12 and add to negative 7 are negative 3 and negative 4, so it factorises to (x - 3)(x - 4) = 0. If two things multiply to give zero then one of them is zero, so either x - 3 = 0, giving x = 3, or x - 4 = 0, giving x = 4. ⚠️ NOTICE THE SIGNS FLIPPED. THE BRACKETS HOLD MINUS THREE AND MINUS FOUR; THE ROOTS ARE PLUS THREE AND PLUS FOUR, and that flip is the single commonest slip in this topic. So check both, separately. Put in 3: nine, minus twenty-one, plus twelve. Nine and twelve is twenty-one, take away twenty-one, and you have zero. Put in 4: sixteen, minus twenty-eight, plus twelve. Sixteen and twelve is twenty-eight, take away twenty-eight, and again zero. ⭐ BOTH ROOTS CONFIRMED, IN UNDER HALF A MINUTE, AND THE SIGN TRAP CANNOT HAVE CAUGHT YOU because a wrong sign would not have produced zero.

Find the larger root

Solve x^2 - 9x + 20 = 0 by factorising. It has two roots. Give the LARGER of them, and check it by substituting it back before you answer.

Order a solve that checks itself

Put the steps in order, from being given an equation to being certain the answer is right.

• Read the equation and decide which kind it is before touching it
• Solve it by the method that kind needs, showing each line of working
• Write the value or values of the unknown down clearly
• Put each value back into the equation you were originally given
• Work out both sides separately and compare them
• If they match, move on; if they do not, go back through the working to find the slip

Assemble the rule for a worded answer

A rectangle problem gives the equation two roots, 7 and negative 2, where the unknown is a length in centimetres. Build the sentence that deals with them correctly.

The checking run

Five questions on solving and on proving your own answer right. Three lives.

Complete the equation solving facts

A value of the unknown that makes an equation true is a _____. To rewrite an expression as a product of brackets is to _____ it. To combine two equations so that one of the unknowns disappears is to _____ it. Moving terms between the two sides, doing the same thing to each side, is to _____ the equation.

Three answers with minutes to spare

Three moments near the end of a paper. In each case the mark is in the reasoning, not the verdict.

• A student has four minutes left. Three questions are answered but unchecked, and one is untouched. What is the best use of the time?
• A student solves a pair of simultaneous equations, checks the pair in the first equation, finds it works, and moves on. What is the risk?
• A student gets an answer they are sure is wrong, rubs out the whole page of working, and writes nothing. How would you advise them?

Explain how to know your answer is right

A classmate solves equations reasonably well but often loses marks to small slips, and never checks anything. Write them an explanation of how to check and why it is worth the time.

• Explain what checking an answer actually means, and why it gives certainty rather than reassurance
• Work through one linear equation of your own, showing both the solving and the check
• Explain what is different about checking a quadratic, and why the signs of the roots are worth watching
• Explain why a pair of simultaneous equations has to be checked in both equations rather than one
• Explain why the check should use the original equation rather than a line you rearranged yourself
• Finish by explaining what to do with working that led to an answer you have decided against