# 9.6.3. Solving Two Equations in Two Variables, One Is of the First Degree, and the Other Is of the Second Degree

A car driver moved a distance of km towards the west, then a distance of km towards the south. If the sum of the two distances is km, and the distance between the starting point and the end point is 35 km, find the distance which the driver moved west and that he moved south.

• Akm, km
• Bkm, km
• Ckm, km
• Dkm, km

### Example

A car driver moved a distance of km towards the west, then a distance of km towards the south. If the sum of the two distances is km, and the distance between the starting point and the end point is 35 km, find the distance which the driver moved west and that he moved south.

### Solution

If the sum of the two distances is km, then

The journey taken by the driver forms a right-angled triangle where the two shorter sides are and and the hypotenuse is the direct distance between the start and end points, which is km. Therefore, these lengths satisfy Pythagoras' theorem.

This pair of equations can be solved by substitution. Rearranging the first order equation gives

Substituting into the second order equation gives

Dividing by 2 gives

This equation can be solved by factorisation.

Now substitute each value of into the first order equation to find . When ,

When ,

Therefore, the driver either moved km west and km south, or km west and km south.

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