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{"name": "Algebraic simultaneous equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Algebraic simultaneous equations", "tags": [], "metadata": {"description": "<p>Students are given 2 equations of the form y=mx+b and asked to solve them using either the substitution or the elimination method. The lines are randomised but the solution coordinates are always integers.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Use the {method} method to find the point of intersection of the following two lines.</p>\n<p>$y = \\var{m1}x \\var{b1sign} \\var{b1}$</p>\n<p>$y= \\var{m2}x \\var{b2sign} \\var{b2}$</p>", "advice": "<p><span data-jme-visible=\"method='substitution'\">To use the&nbsp;substitution method, we set the two expressions that are equal to $y$ equal to each other.</span></p>\n<p><span data-jme-visible=\"method='substitution'\">That is, </span></p>\n<p><span data-jme-visible=\"method='substitution'\">$\\var{m1}x \\var{b1sign}&nbsp;\\var{b1} = \\var{m2}x \\var{b2sign} \\var{b2}$</span></p>\n<p><span data-jme-visible=\"method='substitution'\">Then we subtract $\\var{m2}x$ from both sides:</span></p>\n<p><span data-jme-visible=\"method='substitution'\">$\\var{m1}x - \\var{m2}x \\var{b1sign} \\var{b1} =&nbsp; &nbsp;\\var{b2}$</span></p>\n<p><span data-jme-visible=\"method='substitution'\">In your working it is fine to replace '$--$' with '$+$' and '$+-$' with '$-$' if they ever appear.</span></p>\n<p><span data-jme-visible=\"method='substitution'\">The we&nbsp;subtract&nbsp;$\\var{b1}$ from both sides.&nbsp;</span></p>\n<p><span data-jme-visible=\"method='substitution'\">$\\var{m1}x -\\var{m2}x =&nbsp; \\var{b2} -&nbsp;\\var{b1} $</span></p>\n<p><span data-jme-visible=\"method='substitution'\">Now all the terms with $x$ are on one side, and all the terms without $x$ are on the other side.</span></p>\n<p><span data-jme-visible=\"method='substitution'\">Factorise out the $x$ on the left hand side:</span></p>\n<p><span data-jme-visible=\"method='substitution'\">$(\\var{m1} -\\var{m2})x =&nbsp; \\var{b2} -&nbsp;\\var{b1} $</span></p>\n<p><span data-jme-visible=\"method='substitution'\">Calculate $(\\var{m1} -\\var{m2})$ and $(\\var{b2}&nbsp;- \\var{b1})$</span></p>\n<p><span data-jme-visible=\"method='substitution'\">$(\\var{m1-m2})x =&nbsp; \\var{b2-b1} $</span></p>\n<p><span data-jme-visible=\"method='substitution'\">Finally, divide both sides by $\\var{m1-m2}$ to get</span></p>\n<p><span data-jme-visible=\"method='substitution'\">$x=\\var{(b2-b1)/(m1-m2)}$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">To use the elimination method, let's start by giving the equations names so we can refer to them:</span></p>\n<p><span data-jme-visible=\"method='elimination'\">&nbsp;$ y = \\var{m1}x + \\var{b1} \\qquad \\qquad$ (1)</span></p>\n<p><span data-jme-visible=\"method='elimination'\">&nbsp;$ y =&nbsp;\\var{m2}x + \\var{b2} \\qquad \\qquad$ (2)</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Let's calculate equation (1) - equation (2). This will eliminate the $y$ variable.</span></p>\n<p><span data-jme-visible=\"method='elimination'\">First just write out the subtraction. Make sure to put brackets around the two parts on the right hand side.</span></p>\n<p><span data-jme-visible=\"method='elimination'\">In your working it is fine to replace '$--$' with '$+$' and '$+-$' with '$-$' if they ever appear.</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$ y - y = (\\var{m1}x + \\var{b1}) - (\\var{m2}x + \\var{b2})$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Then rearrange the right hand side:</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$ 0 =&nbsp;\\var{m1}x + \\var{b1} - \\var{m2}x&nbsp;- \\var{b2}$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$ 0 =&nbsp;\\var{m1}x&nbsp;- \\var{m2}x + \\var{b1} -&nbsp;\\var{b2}$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Now calculate $\\var{b1} -&nbsp;\\var{b2}$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$ 0 =&nbsp;\\var{m1}x&nbsp;- \\var{m2}x + \\var{b1-b2}$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Add&nbsp;$\\var{b2-b1}$&nbsp;to both sides of the equation:</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$\\var{b2-b1} =&nbsp;\\var{m1}x&nbsp;- \\var{m2}x$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Factorise the $x$ out on the right hand side.</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$\\var{b2-b1} = (\\var{m1} - \\var{m2})x$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Calculate $&nbsp;&nbsp;(\\var{m1} - \\var{m2})$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$\\var{b2-b1} = (\\var{m1-m2})x$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">Divide both sides by $(\\var{m1-m2})$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">$\\var{(b2-b1)/(m1-m2)} = x$</span></p>\n<p><span data-jme-visible=\"method='elimination'\">So $x=\\var{px}$</span></p>\n<p>Now substitute this value for $x$ back into one of the original equations:</p>\n<p>$y = \\var{m1} \\times \\var{px} \\var{b1sign}&nbsp;\\var{b1}$</p>\n<p>$y = \\var{m1*px+b1}$</p>\n<p>The point of intersection is $(\\var{px},\\var{py})$.</p>\n<p>We can check that this point is correct by substituting it back into both of the original equations and seeing that they are both correct.</p>\n<p></p>", "rulesets": {}, "extensions": [], "variables": {"m1": {"name": "m1", "group": "Unnamed group", "definition": "random(-1,1)*random(1..6)", "description": "", "templateType": "anything"}, "m2temp": {"name": "m2temp", "group": "Unnamed group", "definition": "random(-1,1)*random(1..6)", "description": "", "templateType": "anything"}, "m2": {"name": "m2", "group": "Unnamed group", "definition": "if(m1=m2temp,m2temp*(-1),m2temp)", "description": "", "templateType": "anything"}, "b1": {"name": "b1", "group": "Unnamed group", "definition": "random(-10..10)", "description": "", "templateType": "anything"}, "b2": {"name": "b2", "group": "Unnamed group", 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