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Chapter 7: Problem 10

Solve each system by graphing. Check the coordinates of the intersection point in both equations. \(\left\\{\begin{array}{l}y=x+1 \\ y=3 x-1\end{array}\right.\)

Short Answer

Expert verified
The solution to the system of equations is the point (1,2).

Step by step solution

01

Graph the equations

Begin by graphing each equation on the same set of axes. The first line, \(y=x+1\), is a straight line that passes through the point (0,1) and has a slope of 1 which indicates that the line rises by 1 unit for each unit move to the right. The second line, \(y=3x-1\), passes through the point (0,-1) and has a slope of 3 which indicates that this line rises by 3 units for each unit move to the right.
02

Identify the intersection point

The intersection point of the two lines is the solution to the system. By visually inspecting the graph, we notice that the lines intersect at the point (1,2).
03

Validate the intersection point

To confirm our solution, we substitute the x and y values of the point of intersection into each of the two original equations. Plugging (1,2) into the first equation gives \(2=1+1\), which is a true statement. Plugging (1,2) into the second equation gives \(2=3*1-1\), again a true statement, confirming that the point of intersection is indeed the solution to the system.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities. \(\left\\{\begin{array}{r}2 x+y<4 \\ x-y>4\end{array}\right.\)

A ball is thrown upward and outward from a height of 6 feet. The table shows four measurements indicating the ball's height at various horizontal distances from where it was thrown. The graphing calculator screen displays a quadratic function that models the ball's height, \(y\), in feet, in terms of its horizontal distance, \(x\), in feet. $$ \begin{array}{|c|c|} \hline \begin{array}{c} \boldsymbol{x} \text {, Ball's } \\ \text { Horizontal } \\ \text { Distance } \\ \text { (feet) } \end{array} & \begin{array}{c} \boldsymbol{y}, \\ \text { Ball's } \\ \text { Height } \\ \text { (feet) } \end{array} \\ \hline 0 & 6 \\ \hline 1 & 7.6 \\ \hline 3 & 6 \\ \hline 4 & 2.8 \\ \hline \end{array} $$ $$ \begin{aligned} &\text { [adadReg }\\\ &\begin{aligned} &y=3 \times 2+b x+c \\ &\bar{y}=.8 \\ &b=2.4 \\ &c=6 \end{aligned} \end{aligned} $$ a. Explain why a quadratic function was used to model the data. Why is the value of \(a\) negative? b. Use the graphing calculator screen to express the model in function notation. c. Use the model from part (b) to determine the \(x\)-coordinate of the quadratic function's vertex. Then complete this statement: The maximum height of the ball occurs ____ feet from where it was thrown and the maximum height is ____ feet.

Graph each linear inequality. \(y>\frac{1}{3} x\)

Find the vertex for the parabola whose equation is given by writing the equation in the form \(y=a x^{2}+b x+c\). \(y=(x-4)^{2}+3\)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12\), it's not necessary for me to graph the linear equation \(3 x-4 y=12\) because the inequality contains a \(<\) symbol, in which equality is not included.
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