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Chapter 10: Problem 18

Solve. $$\begin{aligned} &y=x^{2}\\\ &3 x=y+2 \end{aligned}$$

Short Answer

Expert verified
(1, 1) and (2, 4)

Step by step solution

01

Express y in terms of x

Given the first equation: \[ y = x^2 \] We already have y in terms of x.
02

Substitute y into the second equation

Substitute \( y = x^2 \) into the second equation: \[ 3x = y + 2 \rightarrow 3x = x^2 + 2 \]
03

Form a quadratic equation

Rearrange the equation to form a standard quadratic equation: \[ x^2 + 2 - 3x = 0 \rightarrow x^2 - 3x + 2 = 0 \]
04

Solve the quadratic equation

Solve the quadratic equation \( x^2 - 3x + 2 = 0 \) using factorization: \[ x^2 - 3x + 2 = (x - 1)(x - 2) = 0 \]Set each factor equal to zero: \[ x - 1 = 0 \rightarrow x = 1 \] \[ x - 2 = 0 \rightarrow x = 2 \]
05

Find the corresponding y values

Using the values of x, find y: For \( x = 1 \), \( y = 1^2 = 1 \) For \( x = 2 \), \( y = 2^2 = 4 \)
06

Write the solution as pairs

The solutions are the pairs \( (x, y) \): \( (1, 1) \) and \( (2, 4) \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

substitution method
The substitution method involves solving one of the equations in a system of equations for one variable in terms of the others. In this exercise, we start with the two equations:
  • \(y = x^2\)
  • \(3x = y + 2\)
Since the first equation already expresses \(y\) in terms of \(x\), we substitute \(y = x^2\) into the second equation. This substitution simplifies our problem by reducing the number of variables, making it easier to solve the equations step-by-step.
quadratic factorization
After substitution, we obtain a quadratic equation, \(3x = x^2 + 2\). We rearrange it into the standard quadratic form:
  • \(x^2 - 3x + 2 = 0\)
To solve the quadratic equation, we use factorization. Factorization involves writing the quadratic as a product of two binomials. In our case:
  • \(x^2 - 3x + 2 = (x - 1)(x - 2) = 0\)
Setting each factor equal to zero, we find our solutions for \(x\):
  • \(x - 1 = 0 \rightarrow x = 1\)
  • \(x - 2 = 0 \rightarrow x = 2\)
Factorization is particularly useful because it breaks down a complex expression into simpler linear factors.
finding function pairs
Once we have the solutions for \(x\), the next step is to find the corresponding values of \(y\). By substituting \(x = 1\) and \(x = 2\) back into the first equation \(y = x^2\), we get the following pairs:
  • For \(x = 1\), \(y = 1^2 = 1\), leading to the pair \((1, 1)\)
  • For \(x = 2\), \(y = 2^2 = 4\), leading to the pair \((2, 4)\)
These pairs (\(1, 1\) and \(2, 4\)) represent the solutions to the original system of equations. Finding these function pairs is crucial because it shows the relationship between the variables clearly.
system of equations
A system of equations is a collection of two or more equations with the same set of variables. In this case, the system consists of:
  • \(y = x^2\)
  • \(3x = y + 2\)
The goal is to find values for \(x\) and \(y\) that satisfy both equations. We can solve such systems using various methods like substitution, elimination, or graphical methods. For our specific case, substitution was optimal because one equation was already solved for a variable. Solving the system provides us with a complete set of solutions that describe where the equations intersect, leading us to the solutions \((1, 1)\) and \((2, 4)\).

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Most popular questions from this chapter

Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. $$\begin{aligned} &x \geq y^{2}\\\ &x-y \leq 2 \end{aligned}$$

Convert to degree measure. $$\frac{\pi}{3}$$

Find an equation of an ellipse centered at the origin that passes through the points \((1, \sqrt{3 / 2})\) and \((\sqrt{3}, 1 / 2)\)

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { piecewise function } & \text { ellipse }\\\ \text { linear equation } & \text { midpoint } \\ \text { factor } & \text { distance } \\ \text { remainder } & \text { one real-number } \\ \text { solution } & \text { solution } \\ \text { zero } & \text { two different real-number } \\\ x \text { -intercept } & \text { solutions } \\ y \text { -intercept } & \text { two different imaginary- } \\ \text { parabola } & \text { number solutions } \\ \text { circle } & \end{array}$$ A(n) __________________________________ is the set of all points in a plane the sum of whose distances from two fixed points is constant.

Solve. $$\begin{aligned} &2 x y+3 y^{2}=7\\\ &3 x y-2 y^{2}=4 \end{aligned}$$
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