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Imagine a puzzle from 5 years ago: Sister was 5 times older than her brother, and now their ages add up to 34. Find out how old her brother will be 6 years from now!
5 years ago, Sister’s age was 5 times the Age of her brother and the sum of present ages of sister and brother is 34 years. What will be the age of her brother after 6 years?
Let’s define variables:
Let s be the sister’s present age.Let b be the brother’s present age.
Translate the information into equations:
Five years ago, the sister’s age was s-5, and the brother’s age was b-5.
The given statement translates to: s-5 = 5(b-5).
The sum of their present ages is 34: s + b = 34.
Solve the system of equations:
We can rewrite the first equation as s = 5b – 20 and substitute it into the second equation:
(5b-20) + b = 34
Combine like terms: 6b = 54
Divide both sides by 6: b = 9
Find the brother’s age after 6 years:
Add 6 to his current age: 9 + 6 = 15
Therefore, the brother will be 15 years old after 6 years.
Linear Equations in Algebra
What are linear equations?
- Linear equations are equations where the highest power of any variable is 1.
- They represent straight lines when graphed on a coordinate plane.
- They have a constant rate of change, called the slope.
Key terms:
- Variables: Letters that represent unknown values (commonly x and y).
- Coefficients: Numerical factors that multiply the variables.
- Constants: Fixed numerical terms.
- Slope: The rate of change of a line, indicating how much y changes for every unit change in x.
- y-intercept: The point where the line crosses the y-axis.
Forms of linear equations:
-
Slope-intercept form: y = mx + b
- m is the slope, b is the y-intercept.
- Example: y = 2x + 3 (slope = 2, y-intercept = 3)
-
Standard form: Ax + By = C
- A, B, and C are constants.
- Example: 3x + 2y = 6
-
Point-slope form: y – y1 = m(x – x1)
- m is the slope, (x1, y1) is a point on the line.
- Example: y – 4 = -1(x + 2)
Solving linear equations:
- Goal: Isolate the variable to find its value.
- Methods:
- Addition/subtraction
- Multiplication/division
- Substitution (for systems of equations)
Applications:
- Modeling real-world situations (e.g., distance-time relationships, cost-quantity relationships)
- Solving problems in physics, chemistry, economics, and other fields
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