E2.5 - A line and a nonlinear equation

E2.5 - A line and a nonlinear equation

Two simultaneous equations can have more than one solution pair. You will substitute a linear relationship into a nonlinear equation, find every resulting pair, and decide which pairs fit a stated situation.

Substituting a line into a nonlinear equation

Two numbers have product eight, and the second is two greater than the first. Must the numbers be two and four? Negative numbers may offer another answer. To find every possibility, use the relationship rather than guess.

You already know that simultaneous equations must be true for the same values of both unknowns. A linear equation has unknowns only to the first power, with no products of unknowns. A nonlinear equation may contain a square, such as y=x2+1y=x^2+1, or a product, such as xy=8xy=8. Here no powers higher than two are needed.

If y=x+2y=x+2, replace every yy in the other equation by x+2x+2. Equal expressions can replace one another. The resulting equation contains only xx, so quadratic methods become available. Once xx is known, return to the linear equation for yy.

Worked example 1: substitute into a product

Question. Solve y=x+2y=x+2 and xy=8xy=8.

Plan. Substitute x+2x+2 for yy, solve the quadratic, then find the matching yy for each root.

x(x+2)=8replace yx2+2x−8=0expand and subtract eight(x+4)(x−2)=0factorisex=−4 or 2find both rootsy=−2 or 4use y=x+2\begin{aligned} x(x+2)&=8 && \text{replace }y\\ x^2+2x-8&=0 && \text{expand and subtract eight}\\ (x+4)(x-2)&=0 && \text{factorise}\\ x&=-4\text{ or }2 && \text{find both roots}\\ y&=-2\text{ or }4 && \text{use }y=x+2 \end{aligned}

Answer. (x,y)=(−4,−2)(x,y)=(-4,-2) or (2,4)(2,4).

Check. (−4)(−2)=8(-4)(-2)=8 and −2=−4+2-2=-4+2; also 2(4)=82(4)=8 and 4=2+24=2+2. Each pair satisfies both original equations.

Change only the sign of the constant in the linear relationship. Keep the product eight, so you can see how that relationship changes the pairs.

Worked example 2: a changed linear relationship

Question. Solve y=x−2y=x-2 and xy=8xy=8.

Plan. Substitute the changed expression for yy, then recover both pairs.

x(x−2)=8replace yx2−2x−8=0expand and collect(x−4)(x+2)=0factorisex=4 or −2find both rootsy=2 or −4use the same linear equation\begin{aligned} x(x-2)&=8 && \text{replace }y\\ x^2-2x-8&=0 && \text{expand and collect}\\ (x-4)(x+2)&=0 && \text{factorise}\\ x&=4\text{ or }-2 && \text{find both roots}\\ y&=2\text{ or }-4 && \text{use the same linear equation} \end{aligned}

Answer. (x,y)=(4,2)(x,y)=(4,2) or (−2,−4)(-2,-4).

Check. 4(2)=84(2)=8 and 2=4−22=4-2; also (−2)(−4)=8(-2)(-4)=8 and −4=−2−2-4=-2-2.

The notation (x,y)(x,y) is an ordered pair: the first entry is the value of xx, and the second is the value of yy. You do not need a graph to use it. It is a compact way to keep each xx with its own matching yy. The word “line” refers to the linear relationship; its graph will be studied later.

Watch out

Replacing xyxy by x+x−2x+x-2 is wrong. The letters are multiplied, so xy=x(x−2)xy=x(x-2). Put the substituted expression in brackets.

Finding all coordinate pairs

A square of a substituted expression needs brackets too: when y=x+1y=x+1, y2=(x+1)2y^2=(x+1)^2. It does not become x2+1x^2+1, because expanding the square also produces 2x2x.

After solving, make one row per root of the quadratic. Each row is a complete ordered pair, sometimes called a coordinate pair. Listing two possible xx values and two possible yy values without matching them risks creating pairs that solve neither equation.

Worked example 3: a square in the second equation

Question. Solve y=x+1y=x+1 and x2+y2=13x^2+y^2=13.

Plan. Substitute into the square, expand carefully, then match each root with its yy value.

x2+(x+1)2=13replace y22x2+2x−12=0expand and collectx2+x−6=0divide by two(x+3)(x−2)=0factorisex=−3 or 2find both roots\begin{aligned} x^2+(x+1)^2&=13 && \text{replace }y^2\\ 2x^2+2x-12&=0 && \text{expand and collect}\\ x^2+x-6&=0 && \text{divide by two}\\ (x+3)(x-2)&=0 && \text{factorise}\\ x&=-3\text{ or }2 && \text{find both roots} \end{aligned}
xxy=x+1y=x+1Ordered pair
−3-3−2-2(−3,−2)(-3,-2)
2233(2,3)(2,3)

Answer. (−3,−2)(-3,-2) and (2,3)(2,3).

Check. Their squared sums are 9+4=139+4=13 and 4+9=134+9=13. Their differences y−xy-x are both 11, as required.

Change only y=x+1y=x+1 to y=x−1y=x-1. The squared sum is unchanged, but the signs in the expansion and the matching pairs change.

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