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$We want to find the intersection points of the graphs given by the following system of equations: {[y=2x-1],[x^(2)+y^(2)=1]:} One of the intersection points is (0,-1). Find the other intersection point. Your answer must be exact.$

We want to find the intersection points of the graphs given by the following system of equations: $\newline$ $\left\{\begin{array}{l} y=2 x-1 \\ x^{2}+y^{2}=1 \end{array}\right.$ $\newline$ One of the intersection points is $(0,-1)$ . $\newline$ Find the other intersection point. Your answer must be exact.

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Scale drawings: scale factor word problems

Full solution

Q. We want to find the intersection points of the graphs given by the following system of equations:

\newline

\left\{\begin{array}{l} y=2 x-1 \\ x^{2}+y^{2}=1 \end{array}\right.

\newline

One of the intersection points is

(0,-1)

.

\newline

Find the other intersection point. Your answer must be exact.

Given Equations: We are given the system of equations: $\newline$ $1$ . $y = 2x - 1$ $\newline$ $2$ . $x^2 + y^2 = 1$ $\newline$ We already know one intersection point is $(0, -1)$ . To find the other intersection point, we can substitute the expression for $y$ from the first equation into the second equation.
Substitute and Expand: Substitute $y = 2x - 1$ into $x^2 + y^2 = 1$ :
$x^2 + (2x - 1)^2 = 1$
Expand the squared term:
$x^2 + (4x^2 - 4x + 1) = 1$
Combine Like Terms: Combine like terms: $\newline$ $x^2 + 4x^2 - 4x + 1 = 1$ $\newline$ $5x^2 - 4x + 1 = 1$ $\newline$ Subtract $1$ from both sides: $\newline$ $5x^2 - 4x = 0$
Factor and Solve for $x$ : Factor out $x$ : $x(5x - 4) = 0$ Set each factor equal to zero: $x = 0 \text{ or } 5x - 4 = 0$ We already know that $x = 0$ gives us the intersection point $(0, -1)$ , so we need to solve for the other value of $x$ : $5x - 4 = 0$
Find y-coordinate: Add $4$ to both sides: $\newline$ $5x = 4$ $\newline$ Divide by $5$ : $\newline$ $x = \frac{4}{5}$
Calculate $y$ : Now that we have the $x$ -coordinate of the other intersection point, we need to find the corresponding $y$ -coordinate. We can do this by substituting $x = \frac{4}{5}$ into the first equation $y = 2x - 1$ : $\newline$ $y = 2\left(\frac{4}{5}\right) - 1$
Final Intersection Point: Calculate the value of $y$ : $y = \frac{8}{5} - 1$ Convert $1$ to a fraction with a denominator of $5$ : $y = \frac{8}{5} - \frac{5}{5}$ $y = \frac{8 - 5}{5}$ $y = \frac{3}{5}$
Final Intersection Point: Calculate the value of $y$ :
$y = \frac{8}{5} - 1$
Convert $1$ to a fraction with a denominator of $5$ :
$y = \frac{8}{5} - \frac{5}{5}$
$y = \frac{8 - 5}{5}$
$y = \frac{3}{5}$ We have found the $x$ -coordinate and $y$ -coordinate of the other intersection point to be $(\frac{4}{5}, \frac{3}{5})$ . This is the exact value of the other intersection point.

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