Learning Objectives
Connect linear equations, systems, and coordinate graphs.
Choose an efficient method while keeping the requested quantity clear.
Use substitution and graphical interpretation to check an answer.
Prerequisites: the three introductory algebra lessons in this section. These mixed questions review linear topics only. Geometry beyond coordinate lines, nonlinear algebra, trigonometry, probability, and statistics remain for later sections.
1. Recognize the Structure
Problem structure | Useful first step |
|---|---|
One equation, one variable | Distribute, collect, and isolate |
Two simultaneous equations | Choose substitution or elimination |
An inequality with a negative coefficient | Track whether division reverses the sign |
A line through two points | Calculate slope |
An intersection of two lines | Solve their equations as a system |
Check whether a problem asks for x, y, an expression such as 2x + 1, or an ordered pair. A correct intermediate value is not enough if the question asks for something else. Substituting into the original statement is a stronger check than repeating the same algebra.
2. Connect a System to a Graph
Worked Example: Two Cost Models
Two hypothetical services charge A(x) = 5 + 2x and B(x) = 11 + x, where x is a nonnegative number of usage units. Find where their costs match and which is cheaper afterward.
Set 5 + 2x = 11 + x. Subtract x and 5 to obtain x = 6. The common cost is 17.
The graphs intersect at (6, 17). For x > 6, the difference A(x) − B(x) = x − 6 is positive, so B costs less.
The intercepts describe the initial charges, and the slopes describe additional cost per usage unit. These meanings explain why the service with the smaller initial charge can later become more expensive.
3. Use Checks That Address the Error
To check an inequality, test the boundary and one point in the proposed solution region. To check a system, substitute into both equations. To check a line, verify a given point and the required slope. These checks target different types of mistakes.
Common Mistakes
A pair that satisfies just one equation does not solve a system.
A line's slope and intercept have different meanings and often different units.
Do not assume an answer must be a whole number unless the context requires a count.
4. Mixed Practice
1. Solve 5(x − 2) = 3x + 8.
2. Solve 4 − 2x ≤ 10.
3. Solve x + y = 9 and 2x − y = 3.
4. Find a line through (2, 1) and (5, 10), then find its y-intercept.
5. Find the slope of a line perpendicular to 2x + 3y = 12.
6. For which k does kx + 4 = 3x + 4 have infinitely many solutions? What happens when k is different?
7. A rectangle has perimeter 34 cm. Its length is 3 cm greater than its width. Find its dimensions and area.
Worked Solutions
1. Expand to get 5x − 10 = 3x + 8. Hence 2x = 18 and x = 9. Both original sides equal 35.
2. −2x ≤ 6, so division by −2 gives x ≥ −3. At x = −3 the sides are equal; x = 0 also satisfies the inequality.
3. Add the equations to get 3x = 12, so x = 4 and y = 5. Both equations check.
4. The slope is (10 − 1)/(5 − 2) = 3. The equation y − 1 = 3(x − 2) becomes y = 3x − 5. Its y-intercept is (0, −5).
5. Rewrite the original as y = (−2/3)x + 4. The perpendicular slope is 3/2 because (−2/3)(3/2) = −1.
6. k = 3 makes the two sides identical. For k ≠ 3, subtraction gives (k − 3)x = 0, so the unique solution is x = 0.
7. Let width be w and length be w + 3. Then 2w + 2(w + 3) = 34, giving 4w = 28 and w = 7 cm. The length is 10 cm and the area is 70 cm². The perimeter check is 2(7 + 10) = 34 cm.
5. Summary
- Linear equations, systems, and line intersections describe related constraints.
- Choose a method from the structure of the given information.
- Check the final answer in the original equations and interpret its units.
- Use errors to decide which specific skill to revisit before moving on.