Question

How many solutions does the system of linear equations represented in the graph have? Coordinate plane with one line that passes through the points 0 comma 2 and 3 comma 1 and another line that passes through the points 0 comma negative 1 and 3 comma negative 2. One solution at (−1, 0) One solution at (0, −1) Infinitely many solutions No solution

Answer 1

Explanation:

How many solutions does the system of linear equations represented in the graph have?

Coordinate plane with one line that passes through the points 0 comma negative 1 and 1 comma negative 3.

One solution at (−1, 0)

One solution at (0, −1)

No solution

Infinitely many solutions

Answer 2

No solution

Explanation:

In a graphed system of linear equations:

• Intersecting lines: When two lines intersect at a single point, the system of equations has one solution.
  
• Parallel lines: If the lines are parallel and do not intersect, there is no common point between them. Therefore, the system of equations has no solution.
  
• Coincident lines: When the two lines are the same (overlap), this implies that all points on one line are also points on the other. In this case, there are infinitely many solutions.

From inspection of the graph, the lines are parallel. Therefore, the system of equations has no solution.

`\hrulefill`

To find the solution of a system of equations given by description only, find the equations of the lines in slope-intercept form.

`\boxed{\begin{minipage}{6.3 cm}\underline{Slope-intercept form of a linear equation}\\\\$y=mx+b$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $b$ is the $y$-intercept.\\\end{minipage}}`

First, find the slopes of the lines by substituting the given points into the slope formula.

`\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=(y_2-y_1)/(x_2-x_1)$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}`

Given points for line 1:

• (x₁, y₁) = (0, 2)
• (x₂, y₂) = (3, 1)

`\implies \textsf{slope}\:(m)=(y_2-y_1)/(x_2-x_1)=(1-2)/(3-0)=-(1)/(3)`

Given points for line 2:

• (x₁, y₁) = (0, -1)
• (x₂, y₂) = (3, -2)

`\implies \textsf{slope}\:(m)=(y_2-y_1)/(x_2-x_1)=(-2-(-1))/(3-0)=-(1)/(3)`

The y-intercept (b) is the value of y when x = 0. Therefore:

• Line 1 passes through (0, 2), so b = 2.
• Line 2 passes through (0, -1), so b = -1.

Therefore, the equations of the lines are:

`\textsf{Line\;1}: \quad y=-(1)/(3)x+2`

`\textsf{Line\;2}: \quad y=-(1)/(3)x-1`

As the slopes of both lines are the same, but the y-intercepts are different, the lines are parallel.

If two lines are parallel, they will never intersect, and so there is no solution to the given system of equations.