Linear Equations in One Variable - Applications and Word Problems

Understanding Linear Equations: A linear equation in one variable is an equation where the highest power of the variable (usually $x$) is 1. It can be visualized as a balanced see-saw where both sides represent equal values. For example, $ax + b = c$ is a standard form where $a$, $b$, and $c$ are constants.

Solving Strategy: To solve word problems, first identify the unknown quantity and represent it with a variable $x$. Visualize the problem as a translation process where phrases like 'is equal to' or 'gives' represent the $=$ sign, and 'sum' or 'increased by' represent the $+$ sign.

Consecutive Integers: When a problem mentions consecutive integers, imagine a number line where integers follow each other without gaps. We represent them as $x$, $x+1$, $x+2$, etc. For consecutive even or odd integers, the gap is 2 units, so we use $x$, $x+2$, $x+4$.

Age-Related Problems: These problems often involve comparing ages at different points in time. If the current age is $x$, visualize a timeline: $n$ years ago, the age was $(x - n)$, and $n$ years from now (hence), the age will be $(x + n)$. It is often helpful to organize this data in a table.

Two-Digit Numbers: A two-digit number can be visualized in its expanded form using place value. If the digit at the tens place is $x$ and the digit at the units place is $y$, the number is represented as $10x + y$. If the digits are reversed, the new number is $10y + x$.

Geometry Applications: Problems often involve perimeters or angles. Visualize a rectangle where the perimeter is the total length of the boundary, calculated as $2(Length + Breadth)$. For triangles, use the angle sum property where the three interior angles add up to $180^{\circ}$.

Rational Numbers and Fractions: If the numerator of a fraction is $x$ and the denominator is $y$, the fraction is $\frac{x}{y}$. Problems typically describe changes to these parts, such as 'if 2 is added to the numerator,' which is written as $\frac{x + 2}{y}$.

Speed, Distance, and Time: These problems use the relationship $Distance = Speed \times Time$. For boat problems, visualize the boat moving with the current (downstream speed = $Speed_{boat} + Speed_{stream}$) or against the current (upstream speed = $Speed_{boat} - Speed_{stream}$).