91Ó°ÊÓ

Linear equations represent straight lines on a graph and are fundamental to understanding algebra. A linear equation typically looks like this: \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. To solve a linear equation for a specific variable, the goal is to isolate that variable on one side of the equation.

To achieve this, you typically perform operations that keep the equation balanced – whatever you do to one side, you must also do to the other. For instance, if the problem asks to solve for \( y \), you would generally move \( x \)-terms to the opposite side by subtracting or adding them, and then divide by the coefficient of \( y \) to get \( y \) on its own. This method of balancing and isolating the variable allows you to find the 'solution' to the equation – in other words, the value or relationship that makes the equation true.

For example, when you start with \( 10.813x - 17.0y = -45.99 \), the first step is to move all other terms except for the \( y \)-terms, which involves subtracting \( 10.813x \) from both sides and then dividing by \( -17.0 \) to isolate \( y \). Through these steps, you simplify the equation to its very basic form, usually aiming to get to the slope-intercept form which is especially useful in graphing linear equations.

The slope is a measure of how steep a line is and the direction it travels on a graph. In technical terms, it’s defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. The formula to find the slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is \( m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \).

A positive slope means the line is going upwards as you move from left to right, while a negative slope means it’s going downwards. A slope of zero means the line is horizontal, and undefined slope (when the run is zero) indicates a vertical line.

In the slope-intercept form of a linear equation, which is \( y = mx + b \), the slope is represented by \( m \). By isolating the \( y \) variable and simplifying the equation, the coefficient of \( x \) will clearly present the slope of the line. As shown in the simplification step of the given exercise, \( y = 0.64x - 2.71 \), the slope is the coefficient of \( x \), which is 0.64. This means for every one unit the line moves horizontally to the right, it moves 0.64 units up.

In the context of linear equations, the y-intercept is the point where the line crosses the y-axis on a graph. It's the value of \( y \) when \( x \) is zero. In the slope-intercept equation \( y = mx + b \), the y-intercept is represented as \( b \).

To find the y-intercept from a linear equation, you simply need to determine the constant term in the equation when it is in the slope-intercept form. If there are no x-terms in the equation or once you have eliminated the x-term (when \( x=0 \)), the remaining number is the y-intercept.

For example, after you've rearranged the equation from the exercise and simplified it, the equation looks like \( y = 0.64x - 2.71 \). The number \( -2.71 \) is where the line crosses the y-axis. Hence, the y-intercept is \( -2.71 \). It tells us that if we drew this line on a coordinate grid, it would intersect the y-axis below the origin, at the point \( (0, -2.71) \). This is a crucial point in drawing the line accurately on the graph and understanding how the line behaves in relation to the y-axis.