Question

solving a linear-linear equation using a system
what lines would you use to solve
$-3x - 2 = 2x + 8$?
graph the line $\boldsymbol{\text{\boxed{}}}$ for the left side of the equation
graph the line $\boldsymbol{\text{\boxed{}}}$ for the right side of the equation

Explanation:

To solve the equation \(-3x - 2 = 2x + 8\) using a system of linear equations, we need to graph the left - hand side and the right - hand side as separate linear functions.

Step 1: Analyze the left - hand side of the equation

The left - hand side of the equation is \(y=-3x - 2\). This is a linear function in the form \(y = mx + b\), where the slope \(m=-3\) and the \(y\) - intercept \(b = - 2\).

Step 2: Analyze the right - hand side of the equation

The right - hand side of the equation is \(y = 2x+8\). This is also a linear function in the form \(y=mx + b\), where the slope \(m = 2\) and the \(y\) - intercept \(b = 8\).

To solve the equation \(-3x - 2=2x + 8\) graphically, we graph the line \(y=-3x - 2\) for the left - hand side of the equation and the line \(y = 2x + 8\) for the right - hand side of the equation. Then, the \(x\) - coordinate of the point of intersection of the two lines gives the solution to the equation.

For the left - hand side ( \(y=-3x - 2\) ), we can identify the line with a slope of \(- 3\) and a \(y\) - intercept of \(-2\) from the given graph options. For the right - hand side ( \(y = 2x+8\) ), we can identify the line with a slope of \(2\) and a \(y\) - intercept of \(8\) from the given graph options.

If we assume that the lines are labeled based on their equations:

• For the left - hand side (\(-3x - 2\)), we graph the line \(y=-3x - 2\) (let's say this corresponds to one of the lines, for example, if we look at the slopes and intercepts, a line with a negative slope and \(y\) - intercept \(-2\) like line \(C\) or \(D\) in the graph, but more precisely, the function \(y=-3x - 2\)).
• For the right - hand side (\(2x + 8\)), we graph the line \(y = 2x+8\) (a line with a positive slope and \(y\) - intercept \(8\), like line \(A\) or \(B\) in the graph, more precisely the function \(y = 2x+8\)).

If we consider the general form of the lines:

• The left - hand side of the equation \(-3x - 2\) is represented by the line \(y=-3x - 2\).
• The right - hand side of the equation \(2x + 8\) is represented by the line \(y = 2x+8\).

So, to fill in the blanks:

• Graph the line \(y=-3x - 2\) for the left side of the equation.
• Graph the line \(y = 2x+8\) for the right side of the equation.

If we assume that in the dropdowns, the options are the equations of the lines:

• For the left side: \(y=-3x - 2\)
• For the right side: \(y = 2x+8\)

Answer:

To solve the equation \(-3x - 2 = 2x + 8\) using a system of linear equations, we need to graph the left - hand side and the right - hand side as separate linear functions.

Step 1: Analyze the left - hand side of the equation

The left - hand side of the equation is \(y=-3x - 2\). This is a linear function in the form \(y = mx + b\), where the slope \(m=-3\) and the \(y\) - intercept \(b = - 2\).

Step 2: Analyze the right - hand side of the equation

The right - hand side of the equation is \(y = 2x+8\). This is also a linear function in the form \(y=mx + b\), where the slope \(m = 2\) and the \(y\) - intercept \(b = 8\).

To solve the equation \(-3x - 2=2x + 8\) graphically, we graph the line \(y=-3x - 2\) for the left - hand side of the equation and the line \(y = 2x + 8\) for the right - hand side of the equation. Then, the \(x\) - coordinate of the point of intersection of the two lines gives the solution to the equation.

For the left - hand side ( \(y=-3x - 2\) ), we can identify the line with a slope of \(- 3\) and a \(y\) - intercept of \(-2\) from the given graph options. For the right - hand side ( \(y = 2x+8\) ), we can identify the line with a slope of \(2\) and a \(y\) - intercept of \(8\) from the given graph options.

If we assume that the lines are labeled based on their equations:

• For the left - hand side (\(-3x - 2\)), we graph the line \(y=-3x - 2\) (let's say this corresponds to one of the lines, for example, if we look at the slopes and intercepts, a line with a negative slope and \(y\) - intercept \(-2\) like line \(C\) or \(D\) in the graph, but more precisely, the function \(y=-3x - 2\)).
• For the right - hand side (\(2x + 8\)), we graph the line \(y = 2x+8\) (a line with a positive slope and \(y\) - intercept \(8\), like line \(A\) or \(B\) in the graph, more precisely the function \(y = 2x+8\)).

If we consider the general form of the lines:

• The left - hand side of the equation \(-3x - 2\) is represented by the line \(y=-3x - 2\).
• The right - hand side of the equation \(2x + 8\) is represented by the line \(y = 2x+8\).

So, to fill in the blanks:

• Graph the line \(y=-3x - 2\) for the left side of the equation.
• Graph the line \(y = 2x+8\) for the right side of the equation.

If we assume that in the dropdowns, the options are the equations of the lines:

• For the left side: \(y=-3x - 2\)
• For the right side: \(y = 2x+8\)