Learning Objectives
- Find equations of straight lines from gradients, intercepts and points.
- Identify and use the condition for parallel lines.
- Solve simultaneous linear equations by elimination and substitution.
- Solve simultaneous equations graphically using intersections.
- Form simultaneous equations from contextual information.
Key Terms
- Parallel lines
- Lines in the same plane that never meet and have equal gradients.
- Simultaneous equations
- Equations that must be satisfied by the same values of the unknowns.
- Elimination
- Combining equations to remove one unknown.
- Substitution
- Replacing one unknown using an equivalent expression.
- Intersection
- The common point of two graphs and therefore their graphical solution.
- Consistent system
- A pair of equations with at least one common solution.
8th Edition Chapter Map
- Equations of straight lines
- Lines through one or two given points
- Parallel lines
- Elimination and substitution
- Graphical solutions and word problems
Finding An Equation From A Gradient And A Point
Start with y=mx+c. Substitute the known gradient for m, then substitute the coordinates of a point to find c. Finally write the complete equation in its simplest form.
Worked Example: Equation Through A Point
Question: Find the equation of the line with gradient -3 passing through (4,7).
- Write y=-3x+c.
- Substitute x=4,y=7: 7=-12+c.
- c=19.
Answer: y=-3x+19.
Equation Through Two Points
First find the gradient using the two-point formula. Then use either point to determine the intercept. Check the final equation by substituting the second point.
Worked Example: Two Points
Question: Find the equation through (-2,9) and (6,-3).
- m=(-3-9)/(6-(-2))=-12/8=-3/2.
- Use y=-\frac32x+c.
- Substitute (-2,9): 9=3+c, so c=6.
Answer: y=-\frac32x+6.
Parallel Lines
Parallel non-vertical lines have the same gradient but different intercepts. If a line is parallel to 2y=6x-5, first rearrange to y=3x-5/2; the required gradient is 3.
Worked Example: Parallel Line
Question: Find the equation of the line parallel to y=4x-1 and passing through (3,2).
- The gradient is 4.
- Write y=4x+c.
- Substitute (3,2): 2=12+c, so c=-10.
Answer: y=4x-10.
Meaning Of A Simultaneous Solution
A solution pair must satisfy both equations. Graphically it is the intersection of the two lines. Algebraically it is found by reducing two unknowns to one. Always substitute the final pair into both original equations as a check.
Elimination
Make the coefficients of one unknown equal or opposite. Add or subtract entire equations. If an equation is multiplied, every term on both sides must be multiplied.
Worked Example: Elimination
Question: Solve 3x+2y=16 and 5x-2y=8.
- Add the equations: 8x=24.
- x=3.
- Substitute into the first equation: 9+2y=16.
- 2y=7, so y=7/2.
Answer: x=3,\ y=\frac72.
Elimination When Multiplication Is Needed
Worked Example: Create Equal Coefficients
Question: Solve 2x+3y=13 and 5x-2y=4.
- Multiply the first equation by 2: 4x+6y=26.
- Multiply the second by 3: 15x-6y=12.
- Add: 19x=38, so x=2.
- Substitute: 4+3y=13, so y=3.
Answer: (x,y)=(2,3).
Substitution
Use substitution when one equation already gives one variable in terms of the other, or when a variable has coefficient 1. Put the substituted expression in brackets before simplifying.
Worked Example: Substitution
Question: Solve y=2x-5 and 3x+4y=18.
- Replace y: 3x+4(2x-5)=18.
- 11x-20=18, so 11x=38 and x=38/11.
- y=2(38/11)-5=21/11.
Answer: x=\frac{38}{11},\ y=\frac{21}{11}.
Graphical Solutions
Draw both lines on the same axes and read the coordinates of their intersection. A graphical answer is approximate unless the intersection lies exactly on grid lines. Parallel distinct lines have no solution. The same line written in two equivalent forms gives infinitely many solutions.
Constructing Equations From Words
Define the unknowns clearly. Translate each independent fact into an equation. Units must be consistent. In price questions, one equation usually represents a total quantity and another represents a total cost.
Worked Example: Ticket Prices
Question: Three adult tickets and two child tickets cost 46. Two adult tickets and five child tickets cost 49. Find the prices.
- Let adult price be a and child price be c.
- Form 3a+2c=46 and 2a+5c=49.
- Multiply the first by 5 and the second by 2: 15a+10c=230 and 4a+10c=98.
- Subtract: 11a=132, so a=12.
- Then 36+2c=46, giving c=5.
Answer: Adult ticket 12; child ticket 5.
Examination Guidance
- Rearrange line equations into y=mx+c before reading gradient and intercept.
- When solving graphically, show both lines and mark the intersection clearly.
- Show multiplied equations in elimination; this protects method marks.
- Check the final pair in both original equations.
Common Mistakes
- Using the reciprocal of the gradient by reversing numerator and denominator.
- Multiplying only some terms of an equation during elimination.
- Forgetting brackets when substituting a negative or multi-term expression.
- Giving only one unknown when the question asks for both.
Knowledge Check And Practice
1. Find the equation of the line with gradient 5 and y-intercept -7.
2. Find the equation through (1,4) and (5,12).
3. Solve x+y=11 and x-y=3.
4. State the number of solutions for two distinct parallel lines.
5. A line parallel to y=-2x+6 passes through (4,1). Find its equation.
Extended Worked Practice
Solving Simultaneous Equations By Elimination
Question: Solve 3x+2y=19 and 5x-2y=13.
- Add the equations to eliminate y: 8x=32.
- Therefore x=4.
- Substitute into 3x+2y=19: 12+2y=19, so y=3.5.
Answer: (x,y)=(4,3.5).
Finding Intercepts Before Drawing
Question: Find the intercepts of 2x+3y=12.
- For the x-intercept, set y=0: 2x=12, so x=6.
- For the y-intercept, set x=0: 3y=12, so y=4.
Answer: Plot (6,0) and (0,4) and join them with a straight line.
A Sum-And-Difference Problem
Question: Two numbers have sum 46 and difference 12. Find the numbers.
- Let the larger number be x and the smaller be y.
- x+y=46 and x-y=12.
- Add: 2x=58, so x=29. Then y=17.
Answer: 29 and 17.
