🍁One, None, Infinitely Many Solutions Practice

A linear equation is an equation that makes a straight line when graphed. It can be written in the form $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.

An equation has one solution if there is exactly one value of the variable that makes the equation true.

An equation has no solution if there is no value of the variable that can satisfy the equation.

An equation has infinitely many solutions if there are countless values of the variable that satisfy the equation, often because the equation simplifies to a true statement.

Rearranging gives $$2x - 5x = -1 - 5$$ , which simplifies to $$-3x = -6$$ , leading to $$x = 2$$ . Thus, there is one solution.

Subtracting $$3x$$ from both sides gives $$-12 = 1$$ , which is false. Therefore, there is no solution.

Simplifying gives $$x - 4 = x - 4$$ , which is true for all $$x$$ . Thus, there are infinitely many solutions.

Rearranging gives $$7x - 4x = 8 + 1$$ , which simplifies to $$3x = 9$$ , leading to $$x = 3$$ . Thus, there is one solution.

Simplifying gives $$12x - 42 - 3 = 12x - 21$$ , which simplifies to $$-45 = -21$$ , which is false. Therefore, there is no solution.

The slope-intercept form is $$y = mx + b$$ , where $$m$$ is the slope and $$b$$ is the y-intercept.