The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. For example, if the equation is y = 2x + 3, then the slope is 2 and the y-intercept is 3. Hint: Identify the coefficient of x as the slope and the constant term as the y-intercept.

To solve by elimination, manipulate the equations so that adding or subtracting them eliminates one variable. For example, for the system: 2x + 3y = 6 and 4x + 6y = 12, you can multiply the first equation by 2 to align coefficients, giving you 4x + 6y = 12. Subtracting gives 0 = 0, indicating infinitely many solutions. Hint: Look for ways to align coefficients to eliminate a variable.

A system of linear equations has no solution if the equations represent parallel lines, meaning their slopes are equal but their y-intercepts are different. For instance, the equations y = 2x + 1 and y = 2x - 3 have the same slope (2) but different intercepts. Hint: Check if the ratios of the coefficients of x and y are the same while the constant term differs.

The second equation is a multiple of the first (it is obtained by multiplying the first equation by 2), indicating that they represent the same line. Thus, the system has infinitely many solutions. Hint: Check if one equation can be obtained by multiplying or dividing the other.

Two intersecting lines in a system of linear equations represent a unique solution, which is the point where the two lines meet. For example, the equations y = x + 1 and y = -x + 3 intersect at one point. Hint: Graph both equations to visualize the intersection point.

The point-slope form of a linear equation is given by y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line. For example, with a slope of 2 and a point (3, 4), the equation becomes y - 4 = 2(x - 3). Hint: Identify a point on the line and the slope to use this form.

A unique solution implies that the lines represented by the equations have different slopes, meaning they are not parallel. For instance, if one equation has a slope of 1 and another has a slope of -1, they will intersect at one point. Hint: Compare the slopes; if they are equal, the lines are parallel and may not intersect.

The intercept form of a linear equation is expressed as y = mx + b, where m is the slope and b is the y-intercept. For example, the equation y = 3x - 5 indicates a slope of 3 and a y-intercept of -5. Hint: Recognize that the y-intercept is where the line crosses the y-axis.