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Mathematics LibreTexts

2.3: Curve Intersection

Intersection of Lines

Recall that if we want to find the intersection point of two lines, we have two choices:  substitution and elimination.

Example 1: Substitution

Solve

\[ x + 2y = 5,\]

\[4x - 3y = -2.\]

Solutions

We use the first equation to solve for \(x\):

  \[ x = 5 - 2y\]

then we plug  this into the second equation to get

\[\begin{align} 4(5 - 2y) - 3y &= -2 \\ -11y + 20 &= -2 \\ y &= 2 \end{align}\]

and stick this back into the equation for \(x\) to get:

\[x = 5 - 2(2) = 1.\]

Example 2: Elimination

Solve 

\[ 2x + 5y = 19\]

\[ 3x - 5y = -9.\]

We add the equation to get

  \[5x = 10,\; x = 2\]

Hence

\[ 2(2) + 5y = 19\]

\[ y = 3.\]

Intersection of Other Curves

Example 3: Substitution

Find the intersection of the curves

\[x^2 + y^2 = 25 \]

and

\[ y = \dfrac{1}{3}x + 3.\]

We use the method of substitution to arrive at

\[\begin{align} x^2+\left(\dfrac{1}{3}x+3\right)^2&=25 \\ x^2+\dfrac{1}{9}x^2+2x+9&=25 \\ 10x^2+18x-144&=0 \\ 5x^2+9x-72&=0 \\ (5x+24)(x-3)&=0 \end{align}\]

\[ x=-\dfrac{24}{5} \; \text{ or }\; x=3\]

\[y=(\dfrac{1}{3}) \left(\dfrac{-24}{5}\right)+3 \;\;\; \text{or} \;\;\; y=\dfrac{1}{3}(3)+3\]

\[y=-\dfrac{7}{5} \; \text{ or } \; y=4. \]

We get the points 

        \(\left(-\dfrac{24}{5},-\dfrac{7}{5}\right)\) and \((3,4)\).

Example 4: Elimination

\[x^2 + 2y^2 = 18\]

\[ 2x^2 + y^2 = 15\]

We multiply the first equation by 2 and subtract the second equation to get:

\[\begin{align} 3y^2&= 21 \\  y^2&= 7 \\ y &= \sqrt{7} \;\;\; \text{or} \;\;\; y = -\sqrt{7} \end{align}\]

Substituting back into the first equation, we get:

\[x^2 + 2(7) = 18\]

\[x = 2 \;\;\; \text{or} \;\;\; x = -2,\]

hence we get the four points:

   \[ (2,\sqrt{7}), (-2,\sqrt{7}), (2,-\sqrt{7}), (-2,-\sqrt{7}).\]

Example 5: Using a Graphing Calculator

We will use a graphing calculator to find the intersection of

\[y^2+ 16x = 0\]

and

\[y^2+ 9x^2-18x = 18.\]

To find the intersection we just use the intersection function on the graphing calculator.

Example 6

We will use the intercept method to solve

\[(x - 7)(x + 4) = (x + 1)^2.\]

We find the intersection of the two curves

\[y = (x - 7)(x + 4)\]

and 

\[y = (x + 1)^2.\]

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