Video: Estimating Solutions to Simultaneous Equations

Using the graph, determine which of the following is a sensible estimate for the solution to the simultaneous equations 2π₯ + 3π¦ = 20, 4π₯ β 4π¦ = 11. [A] π₯ = 5.4, π¦ = 3.1 [B] π₯ = 5.4, π¦ = 2.9 [C] π₯ = 5.6, π¦ = 2.4 [D] π₯ = 5.7, π¦ = 2.9 [E] π₯ = 5.9, π¦ = 2.7

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Video Transcript

Using the graph, determine which of the following is a sensible estimate for the solution to the simultaneous equations two π₯ plus three π¦ equals 20, four π₯ minus four π¦ equals 11. (a) π₯ equals 5.4 and π¦ equals 3.1. (b) π₯ equals 5.4 and π¦ equals 2.9. (c) π₯ equals 5.6 and π¦ equals 2.4. (d) π₯ equals 5.7 and π¦ equals 2.9. Or (e) π₯ equals 5.9 and π¦ equals 2.7.

Now, It may not be immediately obvious, but the two lines weβve been given on the graph do represent the two equations given in the question. The red line is two π₯ plus three π¦ equals 20, and the blue line is four π₯ minus four π¦ equals 11. The solution to this pair of simultaneous equations then will be the coordinates of the point where these two lines intersect. But from looking at the figure, we can see that they intersect in the middle of one of the smaller squares. So, we canβt find an exact value for the solution. Instead, weβre going to be looking for an estimate.

Letβs first make sure weβre clear on the scale that has been used on each of our axes. In each case, itβs the same. There are four small squares to represent two units. Dividing by four, we see that each small square on each axes represents 0.5 units. Looking at the horizontal placement of this point first of all then, we can see that it is located between the line three small squares to the right of four and then the π₯-value six. If each small square is 0.5, then three small squares is 1.5, which means that the π₯-value to the left of this point is 5.5. And so, our π₯-value is between 5.5 and six.

In the same way, looking at the vertical placement of this point, we can see that itβs located between one small square above two, thatβs 2.5, and two small squares above two, thatβs three. So, the π¦-value is between 2.5 and three.

Looking at the five options, we can rule out options (a) and (b), as their π₯-values are out of range, and option (c), as its π¦-value is out of range. To decide between the two remaining options then, we know that the point is vertically very close to three, and it seems to be closer to 5.5 than to six. So, option (d) is the most sensible estimate. π₯ is approximately equal to 5.7, and π¦ is approximately equal to 2.9.