Linear Equations and Functions
Linear Equations and Functions tests whether you can translate a relationship into an equation, solve it exactly, and read the same relationship from a graph. Expect simultaneous equations, linear functions y=ax+b, intersections, and short word problems about rates or costs.
Core concepts
- Solve equations by doing the same operation to both sides. For simultaneous equations, elimination is usually fastest when coefficients match; substitution is useful when one variable is already isolated.
- In y=ax+b, a is the gradient, or rate of change, and b is the y-intercept. A line rising from left to right has a>0; a horizontal line has a=0.
- The intersection of two lines is the pair (x,y) that satisfies both equations, so solving the system and reading the graph should give the same answer.
- Parallel lines have equal gradients. In word problems, define the variable, write the equation from the stated condition, solve, then answer the actual question.
Worked example
Solve 3x+2y=13 and x−y=1. From the second equation, x=y+1. Substitute into the first: 3(y+1)+2y=13, so 3y+3+2y=13, giving 5y=10 and y=2. Then x=y+1=3. Check: 3(3)+2(2)=13 and 3−2=1. The solution is (x,y)=(3,2).
Traps
- Subtracting equations without changing every sign.
- Finding x and forgetting to find y.
- Mixing up gradient and intercept when a graph is given.
- Giving the equation instead of the requested value in a word problem.
- Assuming two lines meet at the origin without checking.