Question

Write a system of linear equations represent by the graph

inequality 1:

inequality 2:​

Answer 1

This system represents the shaded area in the graph, indicating all the (x, y) points that satisfy both inequalities simultaneously.

To write a system of linear inequalities represented by the graph, we will follow these steps:

1. Identify the equations of the lines.

2. Determine the inequality by looking at the shaded area relative to each line.

3. Write the system of inequalities.

From the image provided, we can see two lines on a coordinate grid. Let's go through each step to create the system of inequalities.

Step 1: Identify the Equations of the Lines

The first line (let's call it Line 1) has a y-intercept at 2 and appears to have a slope that goes down 2 units for every 1 unit it moves to the right. Therefore, the slope of this line is -2, and its equation is
`\( y = -2x + 2 \)`.

The second line (Line 2) has a y-intercept at 1 and a slope that goes up 1 unit for every 4 units it moves to the right. Therefore, the slope of this line is
`\( (1)/(4) \)`, and its equation is
`\( y = (1)/(4)x + 1 \)`.

Step 2: Determine the Inequality

• For Line 1, the shading is above the line. Since the line itself has a negative slope, the inequality will be
  `\( y > -2x + 2 \)` because the area above the line is shaded.
• For Line 2, the shading is below the line. Since the line has a positive slope, the inequality will be
  `\( y < (1)/(4)x + 1 \)` because the area below the line is shaded.

Step 3: Write the System of Inequalities

Now we can write the system of inequalities that represents the shaded region in the graph:

Inequality 1:
`\( y > -2x + 2 \)`

Inequality 2:
`\( y < (1)/(4)x + 1 \)`

Answer 2

Answer: From graph,

inequality 1: y >
`(2)/(3)`x+(-2)

inequality 2: y > (-4)x+2

Explanation:

The graph shows two lines on the X-Y plane.

Step 1: Find the equation of a line.

From figure, One line is passing through (3,0) and (0,-2)

Slope of line is given by m=
`(Y2-Y1)/(X2-X1)`

m=
`((-2)-0)/(0-3)`

m=
`(2)/(3)`

Y-intercept isc=(-2)

The equation of line is given by y=mx+c

Therefore, y=
`(2)/(3)`x+(-2)

From figure, Another line is passing through (0.5,0) and (0,2)

Slope of line is given by m=
`(Y2-Y1)/(X2-X1)`

m=
`(2-0)/(0-0.5)`

m=
`(2)/(-0.5)`

m=(-4)

Y-intercept isc=(2)

The equation of line is given by y=mx+c

Therefore, y=(-4)x+2

Step 2: Test of origin and finding inequality

For y=
`(2)/(3)`x+(-2)

Let suppose, y >
`(2)/(3)`x+(-2)

This line is shaded toward the origin then, inequality must satisfy the origin

Test for origin says,

0 >
`(2)/(3)`(0)+(-2)

0 > (-2)

TRUE.

y >
`(2)/(3)`x+(-2) is required inequality

For y=(-4)x+2

Let suppose, y > (-4)x+2

This line is shaded away from the origin then, inequality must not satisfy the origin

Test for origin says,

0 >
`(2)/(3)`(0)+(-2)

0 > (+2)

FALSE, so that y > (-4)x+2 does not satisfy the origin

y > (-4)x+2 is required inequality

Thus,

inequality 1: y >
`(2)/(3)`x+(-2)

inequality 2: y > (-4)x+2