Write each equation in slope-intercept form. Without...

Write each equation in slope-intercept form. Without graphing the equations, state whether the system of equations has exactly one solution, no solution, or an infinite number of solutions. $x+y=-2$
$3 y+12=-6 x$

Key Concepts

Slope

The slope of a line indicates the rate of change of the y-coordinate with respect to the x-coordinate and is a measure of the steepness of the line. It is a critical concept in analyzing linear equations because, by comparing slopes, one can determine whether lines are parallel (equal slopes) or intersecting (different slopes), which in turn informs the nature of the solutions in a system of equations.

Types of Solutions in Linear Systems

When dealing with systems of linear equations, there can be exactly one solution (when the lines intersect at a single point), no solution (when the lines are parallel and never intersect), or an infinite number of solutions (when the lines are coincident, meaning they lie on top of each other). The nature of the solution is typically revealed by comparing the slopes and intercepts after converting the equations into a common form.

Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same set of variables. The solution to the system is the set of values that satisfy all the equations simultaneously. Methods of solving these systems include substitution, elimination, and comparing slopes (if transformed into slope-intercept form), allowing one to determine whether the system has a unique solution, no solution, or infinitely many solutions.

Slope-Intercept Form

This is a way of expressing the equation of a line so that its slope and y-intercept are immediately visible. In this form, an equation is written as y = mx + b, where m represents the slope of the line and b represents the point at which the line crosses the y-axis. It is a standard form used for graphing linear equations and comparing the slopes and intercepts of different lines.

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