Now that we have analyzed part a and b, we will finish up with part c; solve real-world and mathematical problems leading to two linear equations in two variables.

The example given under part c of Standard 10 says, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. Let us test our knowledge and see if we can determine if the line that passes through (2,7) and (1,4) intersects the line that passes through (4,9) and (0,5).

Using Desmos, I graph the given points.

For our first pair, (2,7) and (1,4), we must find the equation of the line.

Using the formula, (y₂-y₁)/(x₂-x₁) we find that the slope of the line that goes through (2,7) and (1,4) is 3. Using the slope, we can determine the equation of the line using the point slope formula, y-y₁=m(x-x₁).

Therefore, the equation of the line that passes through (2,7) and (1,4) is y=3x+1.

Now let’s find the equation of the line that passes through (4,9) and (0,5). Completing the same steps above, determine the slope.

Using the same formula, (y₂-y₁)/(x₂-x₁), we find that the slope of the line is 1. With the slope, we can determine the equation of the line using the point slope formula, y-y₁=m(x-x₁).

Therefore, the equation of the line that passes through (4,9) and (0,5) is y=x+5.

However, part c of standard #10 asks us to determine whether the line through the first pair of points intersects the line through the second pair. This is where we will apply our knowledge from part a and b.

Similar to what we did in part b, to determine if the lines intersect, we must set the equations found above equal to each other. Setting 3x+1 equal to x+5 and solving for x results in an x-value of 2.

Once we know the x-value, we substitute it back in to either equation to solve for the y-value. Substituting x=2 into either equation gives us a y-value of 7.

Therefore, using what we discussed in part a, solutions to a system of two linear equations in two variables correspond to points of intersections of their graphs, we have determined that the lines passing through either (2,7) and (1,4) or (4,9) and (0,5) intersect at point (2,7) because this is the point that simultaneously satisfies both linear equations.

Next, I will discuss you might use or adapt this activity for the mathematics classroom.