Solid vs Dashed Lines and Test Points

Worked example

1. 1

   $x=2.5$ — Every point on it has x equal to 2.5, so it is an upright line through 2.5 on the x-axis, halfway between 2 and 3.

2. 2

   $y=x$ — y always equals x, so it passes through (0, 0), (1, 1) and (2, 2). Join them: it goes up one square for every square across.

3. 3

   $x+y=8$ — The two coordinates always add to 8, like (0, 8), (5, 3) and (8, 0). The next card shows the quick way to draw it.

Worked example

1. 1

   $x+y=8$ — The line. Find two easy points on it.

2. 2

   $x=0:0+y=8,y=8$ — On the y-axis x is 0, so y must be 8. That gives the point (0, 8).

3. 3

   $y=0:x+0=8,x=8$ — On the x-axis y is 0, so x must be 8. That gives the point (8, 0).

4. 4

   $(5,3):5+3=8✓$ — Check a third point before drawing. It adds to 8, so it is on the line. Now join (0, 8) and (8, 0).

Worked example

1. 1

   $2x+3y=12$ — The line. Here a = 2, b = 3 and c = 12.

2. 2

   $x=0:3y=12$ — On the y-axis x is 0, so the 2x term becomes 0 and drops out.

3. 3

   $y=\frac{12}{3}=4$ — Divide both sides by 3. The line crosses the y-axis at (0, 4).

4. 4

   $y=0:2x=12$ — On the x-axis y is 0, so the 3y term drops out.

5. 5

   $x=\frac{12}{2}=6$ — Divide both sides by 2. The line crosses the x-axis at (6, 0).

6. 6

   $(3,2):2(3)+3(2)=6+6=12✓$ — Check a third point: it gives 12, so it is on the line. Join (0, 4) and (6, 0) with a ruler.

Worked example

1. 1

   $x\ge 2.5,y\ge x,x+y\le 8$ — The three inequalities. R is the triangle they leave.

2. 2

   $(3,4)$ — Pick a point inside the triangle, not on any line.

3. 3

   $x\ge 2.5:3\ge 2.5✓$ — Put x = 3 into the first one. True.

4. 4

   $y\ge x:4\ge 3✓$ — Put y = 4 and x = 3 into the second. True.

5. 5

   $x+y\le 8:3+4=7\le 8✓$ — Add the coordinates for the third: 7 is at most 8, so true. All three hold, so (3, 4) is in R.

6. 6

   $(4,3):3\ge 4\times$ — Now a point below y = x. It fails the second test, and one false is enough: it is outside R.