16. Linear equations and graphs
Free study material · concepts, shortcuts & solved questions
Transpose terms carefully: ax+b=c gives x=(c−b)/a. For simultaneous equations, eliminate or substitute one variable. A linear equation in two variables represents a straight line; intercepts can be found by setting the other variable to zero.
Method before formula Always substitute the final answer back. In graph questions, identify whether the line has positive or negative slope and whether it crosses the axes where the options suggest. |
Formula and decision table
Example | Result |
3x+7=25 | x=6 |
2x+y=7, x−y=2 | x=3, y=1 |
y=2x+1 | Slope=2, y-intercept=1 |
Worked understanding
A reliable solution has five visible moves: identify the data, choose the base or unit, write the rule, substitute carefully, and check the size of the answer. The examples in this chapter are deliberately small so that the method remains visible.
Calculation discipline Before pressing ahead, state the denominator or unit in words. For example: profit on CP, discount on MP, average over number of items, speed in metres per second, and probability over the fixed sample space. |
Solved examples from this chapter
Question focus | Correct move | Answer |
For 3x+7=25, x: | 3x=18, so x=6. | C. 6 |
The graph of y=2x+1 has slope: | The coefficient of x is the slope. | D. 2 |
In elimination, equations are combined to remove: | The goal is one equation in one variable. | A. A variable |
What an examiner is testing
The options around Linear equations and graphs usually represent a wrong base, a missed conversion, a sign error or a shortcut used outside its condition. Say the base and unit before calculating, estimate the result, and use substitution or a reverse operation as the final check.
Step-by-step answer construction
Example 1: For 3x+7=25, x: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 6, because 3x=18, so x=6.
Example 2: The graph of y=2x+1 has slope: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is 2, because The coefficient of x is the slope.
Example 3: In elimination, equations are combined to remove: First identify the requested quantity and its base or unit. Apply the chapter rule, keep the operation visible, estimate the expected range, and compare the result with the options. The correct answer is A variable, because The goal is one equation in one variable.
Chapter practice
1. For 3x+7=25, x:
(A) 4 (B) 5 (C) 6 (D) 7
Answer: C. 6 | Explanation: 3x=18, so x=6.
2. The graph of y=2x+1 has slope:
(A) −2 (B) −1 (C) 1 (D) 2
Answer: D. 2 | Explanation: The coefficient of x is the slope.
3. In elimination, equations are combined to remove:
(A) A variable (B) All constants (C) The graph (D) Units
Answer: A. A variable | Explanation: The goal is one equation in one variable.
SSC CGL speed and trap clinic
For Linear equations and graphs, speed comes after classification. Before calculating, say aloud what the number means, what the denominator is, and what unit the answer must carry. Then use the nearest-option check to catch a sign, base or conversion error.
Checkpoint | What to verify | Typical SSC mistake |
Base | Which quantity is the base in Linear equations and graphs? | Using selling price instead of cost price, or part instead of total |
Unit | Are time, distance, area and rate in compatible units? | Mixing hours with minutes or km/h with m/s |
Direction | Should the answer increase, decrease or stay bounded? | Accepting an impossible negative length or probability above 1 |
Estimate | What range should the answer lie in before exact work? | Trusting a long calculation that is far from the options |
Reverse check | Can the answer be substituted back into the condition? | Stopping at an algebraic value without verification |
Mini decision drill
1. Write the first operation you would perform in a Linear equations and graphs question and why.
2. State the most dangerous denominator or unit in this chapter.
3. Give one condition under which a shortcut would be invalid.
4. Create a small numerical example and verify it by a second method.
Revision evidence Do not tick this chapter because you recognised the formula. Tick it only after you solve one direct question, one altered-condition question and one mixed-paper question correctly. |