June 2023 Paper 1 Q18
18 Solve the simultaneous equations
\[\begin{aligned} 2x + y &= 13 \\ x + 3z &= 2 \\ z - 2y &= -7 \end{aligned}\]Do not use trial and improvement.
You must show your working. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative Method 1 – Elimination of \(x\) from the first 2 equations | ||
| Correctly setting up two equations to eliminate \(x\) with an attempt to add or subtract from two equations | M1 | eg \(2(x + 3z) - (2x + y)\) |
| Correct method and terms to eliminate a second variable | M1dep | eg \((6z - y) - 6(z - 2y)\) |
| Correctly eliminating a variable from their two correct equations to form an equation in one variable | M1dep | eg \(y - 12y = 9 - 42\) or \(11y = 33\) or \(y = 3\) |
| Two correct values with working seen | A1 | eg \(y = 3\) and \(x = 5\) |
| \(x = 5\) and \(y = 3\) and \(z = -1\) with correct working seen | A1 | oe |
Additional guidance
| For first two M marks LHS manipulation only required | M2 |
| All three M marks can be combined in one equation: ie \(2(x + 3z) - (2x + y) - 6(z - 2y) = 11y\) and \(11y = 33\) and \(y = 3\) | M3 |
| May be multiple attempts. M marks can be awarded for best attempt but A marks would have to be from answers chosen | |
| Stop marking after first error on LHS (but check for restarts) | |
| Stop marking as soon as T&I/inspection methods begin (but check for restarts) |
| Answer | Mark | Comments |
|---|---|---|
| Alternative Method 2 – Elimination of \(y\) from 2 equations | ||
| Correctly setting up two equations to eliminate \(y\) with an attempt to add or subtract from two equations | M1 | eg \(2(2x + y) + (z - 2y)\) |
| Correct method and terms to eliminate a second variable | M1dep | eg \((4x + z) - 4(x + 3z)\) |
| Correctly eliminating a variable from their two correct equations to form an equation in one variable | M1dep | eg \(z - 12z = -7 + 18\) or \(-11z = 11\) or \(z = -1\) |
| Two correct values with working seen | A1 | eg \(z = -1\) and \(x = 5\) |
| \(x = 5\) and \(y = 3\) and \(z = -1\) with correct working seen | A1 | oe |
Additional guidance
| For first two M marks LHS manipulation only required | M2 |
| All three M marks can be combined in one equation: ie \(2(x + 3z) - (2x + y) - 6(z - 2y) = 11y\) and \(11y = 33\) and \(y = 3\) | M3 |
| May be multiple attempts. M marks can be awarded for best attempt but A marks would have to be from answers chosen | |
| Stop marking after first error on LHS (but check for restarts) | |
| Stop marking as soon as T&I/inspection methods begin (but check for restarts) |
| Answer | Mark | Comments |
|---|---|---|
| Alternative Method 3 – Elimination of \(z\) from 2 equations | ||
| Correctly setting up two equations to eliminate \(z\) with an attempt to add or subtract from two equations | M1 | eg \((x + 3z) - 3(z - 2y)\) |
| Correct method and terms to eliminate a second variable | M1dep | eg \(2(x + 6y) - (2x + y)\) |
| Correctly eliminating a variable from their two correct equations to form an equation in one variable | M1dep | eg \(y - 12y = 9 - 42\) or \(11y = 33\) or \(y = 3\) |
| Two correct values with working seen | A1 | eg \(y = 3\) and \(x = 5\) |
| \(x = 5\) and \(y = 3\) and \(z = -1\) with correct working seen | A1 | oe |
Additional guidance
| For first two M marks LHS manipulation only required | M2 |
| All three M marks can be combined in one equation: ie \(2(x + 3z) - (2x + y) - 6(z - 2y) = 11y\) and \(11y = 33\) and \(y = 3\) | M3 |
| May be multiple attempts. M marks can be awarded for best attempt but A marks would have to be from answers chosen | |
| Stop marking after first error on LHS (but check for restarts) | |
| Stop marking as soon as T&I/inspection methods begin (but check for restarts) |
| Answer | Mark | Comments |
|---|---|---|
| Alternative Method 4 – Substitution | ||
| One equation rearranged and correctly substituted into a second equation to eliminate an unknown | M1 | eg \(x = 2 - 3z\) and \(2(2 - 3z) + y\) |
| Two correct equations formed with the same two unknowns in each equation (must be simplified correctly) and one equation rearranged so it can be substituted into the second equation | M1dep | eg \(-6z + y = 9\) and \(z = 2y - 7\) |
| Correctly eliminating a variable from their two correct equations to form an equation in one variable | M1dep | eg \(-6(2y - 7) + y = 9\) or \(-11y = -33\) or \(y = 3\) |
| Two correct values with working seen | A1 | eg \(z = -1\) and \(y = 3\) |
| \(x = 5\) and \(y = 3\) and \(z = -1\) with correct working seen | A1 | oe |
Additional guidance
| Could use a combination of elimination and substitution at any point. Stop marking after first error | |
| May be multiple attempts. M marks can be awarded for best attempt but A marks would have to be from answers chosen | |
| Stop marking as soon as T&I/inspection methods begin (but check for restarts) |