Video: Determining the Area of a Triangle Bounded by the π‘₯𝑦-Axes and a Given Line

Determine the area of the triangle bounded by the π‘₯-axis, the 𝑦-axis, and the straight line 2π‘₯ + 7𝑦 + 28 = 0.

03:08

Video Transcript

Determine the area of the triangle bounded by the π‘₯-axis, the 𝑦-axis, and the straight line two π‘₯ plus seven 𝑦 plus 28 equals zero.

Let’s begin by drawing the line two π‘₯ plus seven 𝑦 plus 28 equals zero. We can create a table of values to find the coordinates where this line will cross both axes. It may be helpful to put this equation into the form 𝑦 equals π‘šπ‘₯ plus 𝑐. Rearranging to have just seven 𝑦 on one side means that we must subtract two π‘₯ and subtract 28 from both sides of the equation, which gives us seven 𝑦 equals negative two π‘₯ minus 28. Dividing through by seven then, we have 𝑦 equals negative two-sevenths π‘₯ minus 28 over seven. And we can notice that this final fraction will simplify to four.

This form of the equation will allow us to more easily find the points where the line crosses the π‘₯- and 𝑦-axes. To determine the point where a line crosses the 𝑦-axis, this could be determined when the π‘₯-value of the coordinate is equal to zero. So substituting π‘₯ equals zero into our equation 𝑦 equals negative two-sevenths π‘₯ minus four gives us 𝑦 equals negative two-sevenths times zero minus four, which simplifies to give us 𝑦 equals negative four.

Recall that, in the general form of an equation 𝑦 equals π‘šπ‘₯ plus 𝑐, the 𝑐-value relates to the 𝑦-intercept. Here, this will be equal to negative four. The coordinate where the line crosses the π‘₯-axis can be determined by finding the value when 𝑦 is equal to zero. So substituting 𝑦 equals zero into our equation, we have zero equals negative two-sevenths π‘₯ minus four.

Rearranging to get π‘₯ by itself, we can begin by adding four to both sides, which gives four equals negative two-sevenths π‘₯. We can then multiply through by seven to give us 28 equals negative two π‘₯. Finally, dividing both sides of our equation by negative two, we have 28 divided by negative two equals π‘₯. So π‘₯ equals negative 14. We can now plot these two coordinates on our equation two π‘₯ plus seven 𝑦 plus 28 equals zero.

We’re now asked to find the area of the triangle bounded by the π‘₯-axis, the 𝑦-axis, and the line two π‘₯ plus seven 𝑦 plus 28 equals zero. So we need to use the formula for the area of a triangle, which is that the area of a triangle is equal to half times the base times the height. So to find the area of our triangle, we could take the base to be the length along the π‘₯-axis, which is 14 units long. The height of the triangle can be taken as four units long. Therefore, we multiply a half, 14, and four, which gives us 28.

Therefore, our final answer for the area of the triangle bounded by the π‘₯-axis, the 𝑦-axis, and the line with equation two π‘₯ plus seven 𝑦 plus 28 equals zero is 28 area units.

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