Solving One Linear and One Non-Linear Simultaneous Equation

When you've got simultaneous equations where one is linear (straight line) and one is non-linear (like a circle or parabola), substitution is your best mate. The process is straightforward: rearrange the linear equation to make one variable the subject, then substitute this into the non-linear equation.

Let's look at the example: x² + y² = 25 (this is a circle) and x + y = 7 (this is a straight line). First, rearrange the linear equation to get y = 7 - x. This gives you y in terms of x.

Next, substitute this expression into the non-linear equation: x² + $7-x$² = 25. Expand the brackets carefully: x² + 49 - 14x + x² = 25. Simplify to get 2x² - 14x + 24 = 0.

Now solve this quadratic using the formula or factoring. You'll get x = 4 and x = 3. Substitute back into y = 7 - x to find the corresponding y-values: when x = 4, y = 3, and when x = 3, y = 4. Your solutions are (3,4) and (4,3) - these are the points where the circle and line intersect.

Quick Tip: Always check your answers by substituting both coordinates back into both original equations!