Q41. - Algebra 1

Question

MULTIPLE REPRESENTATIONS In this problem, you will explore $\mathbf{2}\mathbf{x}+\mathbf{4}=-\mathbf{x}-\mathbf{2}$.

a. GRAPHICAL Make a table of values with five points for $\mathbf{y}=\mathbf{2}\mathbf{x}+\mathbf{4}$ and $\mathbf{y}=-\mathbf{x}-\mathbf{2}$. Graph the points from the tables.

b. ALGEBRAIC Solve $\mathbf{2}\mathbf{x}+\mathbf{4}=-\mathbf{x}-\mathbf{2}$

c. VERBAL Explain how the solution you found in part b is related to the intersection point of the graphs in part a.

Step-by-Step Solution

Answer

a. The graph obtained is:

b. The solution of the given equation is $x=-2$

c. The solution obtained in part (b) represents the point of intersection of the curves obtained in part (a).

Part a. Step 1. State the concept.

A function can be graphed by joining multiple points lying on the curve of the function. These points can be obtained by putting random values of $x$ and obtaining the respective values for $y$.

Part a. Step 2. State the objective.

The objective is to make a table of values with five points for the functions $y=2x+4$ and $y=-x-2$, and to graph these points.

Part a. Step 3. Make the table.

Given functions: $y=2x+4$ and $y=-x-2$

 

The required table of points is:

 

x	$y=2x+4$	$y=-x-2$
$-2$	0	0
$-1$	2	$-1$
0	4	$-2$
1	6	$-3$
2	8	$-4$

 

Part a. Step 4. Graph the points

The graph of the points in the above table is as follows:

In the above graph, the green crosses represent the points on the curve of $y=2x+4$ and the violet dots represent the points on the curve of $y=-x-2$.

Part b. Step 1. Write the given equation.

The given equation is “$2x+4=-x-2$”.

Part b. Step 2. Solve the given equation.

Solving,

 

$2x+4=-x-2$  (Given equation)

 

$2x+4+x=-x-2+x$  (Add  to both sides)

 

$3x+4=-2$  (Simplify)

 

$3x+4-4=-2-4$  (Subtract 4 from both sides)

 

$3x=-6$  (Simplify)

 

$\frac{3x}{3}=\frac{-6}{3}$  (Divide both sides by 3)

 

$x=-2$  (Simplify)

 

So, $x=-2$ is the solution of the given equation.

Part c. Step 1. State the concept.

The solution of two linear equations is nothing but the point where their corresponding curves intersect.

Part c. Step 2. Write the objective.

The objective is to explain how the solution obtained in part b is related to the intersection point of the graphs in part a.

Part c. Step 3. Explanation.

As seen from the graph in part (a), $\left(-2,0\right)$ is an intersection point of both curves. Also, it was obtained in part (b), that the solution of the two linear equations is $x=-2$. Clearly, the solution represents the point of intersection of the curves.