Solved on Jan 30, 2024

Find the intersection point of the linear equations $y = -6x + 2$ and $y = 3x - 7$ .

STEP 1

Assumptions
1. We have two linear equations in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We want to find the intersection point of the two lines represented by these equations.

STEP 2

To graph the first equation $y = -6x + 2$ , we need to identify two points that lie on this line. We can do this by choosing two different values for $x$ and calculating the corresponding $y$ values.

STEP 3

Let's choose $x = 0$ and $x = 1$ for the first equation.
For $x = 0$ : $y = -6(0) + 2 = 2$
For $x = 1$ : $y = -6(1) + 2 = -6 + 2 = -4$

STEP 4

Now we have two points for the first line: (0, 2) and (1, -4). We can plot these points on a graph and draw a straight line through them to represent the first equation.

STEP 5

Next, we graph the second equation $y = 3x - 7$ in the same way.

STEP 6

Let's choose $x = 0$ and $x = 1$ for the second equation.
For $x = 0$ : $y = 3(0) - 7 = -7$
For $x = 1$ : $y = 3(1) - 7 = 3 - 7 = -4$

STEP 7

Now we have two points for the second line: (0, -7) and (1, -4). We can plot these points on the same graph as the first line and draw a straight line through them to represent the second equation.

STEP 8

After plotting both lines on the graph, we look for the point where the two lines intersect. This point is the solution to the system of equations, which means it satisfies both equations simultaneously.

STEP 9

By observing the graph, we can determine the coordinates of the intersection point. If the graph is not precise enough, we can find the intersection algebraically by setting the two equations equal to each other and solving for $x$ .
$-6x + 2 = 3x - 7$

STEP 10

Combine like terms by moving all terms involving $x$ to one side and constant terms to the other side.
$-6x - 3x = -7 - 2$

STEP 11

Simplify the equation by combining the terms.
$-9x = -9$

STEP 12

Divide both sides by -9 to solve for $x$ .
$x = \frac{-9}{-9} = 1$

STEP 13

Now that we have the value of $x$ , we can substitute it back into either of the original equations to find the corresponding value of $y$ . Let's use the first equation $y = -6x + 2$ .
$y = -6(1) + 2$

STEP 14

Calculate the value of $y$ .
$y = -6 + 2 = -4$

STEP 15

We have found the intersection point by solving the equations algebraically. The intersection point is $(1, -4)$ .
The intersection point of the two lines represented by the equations $y = -6x + 2$ and $y = 3x - 7$ is $(1, -4)$ .

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Find the intersection point of the linear equations $y = -6x + 2$ and $y = 3x - 7$ .

STEP 1

Assumptions
1. We have two linear equations in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We want to find the intersection point of the two lines represented by these equations.

STEP 2

To graph the first equation $y = -6x + 2$ , we need to identify two points that lie on this line. We can do this by choosing two different values for $x$ and calculating the corresponding $y$ values.

STEP 3

Let's choose $x = 0$ and $x = 1$ for the first equation.
For $x = 0$ : $y = -6(0) + 2 = 2$
For $x = 1$ : $y = -6(1) + 2 = -6 + 2 = -4$

STEP 4

Now we have two points for the first line: (0, 2) and (1, -4). We can plot these points on a graph and draw a straight line through them to represent the first equation.

STEP 5

Next, we graph the second equation $y = 3x - 7$ in the same way.

STEP 6

Let's choose $x = 0$ and $x = 1$ for the second equation.
For $x = 0$ : $y = 3(0) - 7 = -7$
For $x = 1$ : $y = 3(1) - 7 = 3 - 7 = -4$

STEP 7

Now we have two points for the second line: (0, -7) and (1, -4). We can plot these points on the same graph as the first line and draw a straight line through them to represent the second equation.

STEP 8

After plotting both lines on the graph, we look for the point where the two lines intersect. This point is the solution to the system of equations, which means it satisfies both equations simultaneously.

STEP 9

By observing the graph, we can determine the coordinates of the intersection point. If the graph is not precise enough, we can find the intersection algebraically by setting the two equations equal to each other and solving for $x$ .
$-6x + 2 = 3x - 7$

STEP 10

Combine like terms by moving all terms involving $x$ to one side and constant terms to the other side.
$-6x - 3x = -7 - 2$

STEP 11

Simplify the equation by combining the terms.
$-9x = -9$

STEP 12

Divide both sides by -9 to solve for $x$ .
$x = \frac{-9}{-9} = 1$

STEP 13

Now that we have the value of $x$ , we can substitute it back into either of the original equations to find the corresponding value of $y$ . Let's use the first equation $y = -6x + 2$ .
$y = -6(1) + 2$

STEP 14

Calculate the value of $y$ .
$y = -6 + 2 = -4$

STEP 15

We have found the intersection point by solving the equations algebraically. The intersection point is $(1, -4)$ .
The intersection point of the two lines represented by the equations $y = -6x + 2$ and $y = 3x - 7$ is $(1, -4)$ .

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Find the intersection point of the linear equations y=−6x+2y = -6x + 2y=−6x+2 and y=3x−7y = 3x - 7y=3x−7.

STEP 1

Assumptions1. We have two linear equations in the form of y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.2. We want to find the intersection point of the two lines represented by these equations.

STEP 2

To graph the first equation y=−6x+2y = -6x + 2y=−6x+2, we need to identify two points that lie on this line. We can do this by choosing two different values for xxx and calculating the corresponding yyy values.

STEP 3

Let's choose x=0x = 0x=0 and x=1x = 1x=1 for the first equation.For x=0x = 0x=0: y=−6(0)+2=2y = -6(0) + 2 = 2y=−6(0)+2=2For x=1x = 1x=1: y=−6(1)+2=−6+2=−4y = -6(1) + 2 = -6 + 2 = -4y=−6(1)+2=−6+2=−4

STEP 4

Now we have two points for the first line: (0, 2) and (1, -4). We can plot these points on a graph and draw a straight line through them to represent the first equation.

STEP 5

Next, we graph the second equation y=3x−7y = 3x - 7y=3x−7 in the same way.

STEP 6

Let's choose x=0x = 0x=0 and x=1x = 1x=1 for the second equation.For x=0x = 0x=0: y=3(0)−7=−7y = 3(0) - 7 = -7y=3(0)−7=−7For x=1x = 1x=1: y=3(1)−7=3−7=−4y = 3(1) - 7 = 3 - 7 = -4y=3(1)−7=3−7=−4

STEP 7

Now we have two points for the second line: (0, -7) and (1, -4). We can plot these points on the same graph as the first line and draw a straight line through them to represent the second equation.

STEP 8

After plotting both lines on the graph, we look for the point where the two lines intersect. This point is the solution to the system of equations, which means it satisfies both equations simultaneously.

STEP 9

By observing the graph, we can determine the coordinates of the intersection point. If the graph is not precise enough, we can find the intersection algebraically by setting the two equations equal to each other and solving for xxx.−6x+2=3x−7-6x + 2 = 3x - 7−6x+2=3x−7

STEP 10

Combine like terms by moving all terms involving xxx to one side and constant terms to the other side.−6x−3x=−7−2-6x - 3x = -7 - 2−6x−3x=−7−2

STEP 11

Simplify the equation by combining the terms.−9x=−9-9x = -9−9x=−9

STEP 12

Divide both sides by -9 to solve for xxx.x=−9−9=1x = \frac{-9}{-9} = 1x=−9−9​=1

STEP 13

Now that we have the value of xxx, we can substitute it back into either of the original equations to find the corresponding value of yyy. Let's use the first equation y=−6x+2y = -6x + 2y=−6x+2.y=−6(1)+2y = -6(1) + 2y=−6(1)+2

STEP 14

Calculate the value of yyy.y=−6+2=−4y = -6 + 2 = -4y=−6+2=−4

STEP 15

We have found the intersection point by solving the equations algebraically. The intersection point is (1,−4)(1, -4)(1,−4).The intersection point of the two lines represented by the equations y=−6x+2y = -6x + 2y=−6x+2 and y=3x−7y = 3x - 7y=3x−7 is (1,−4)(1, -4)(1,−4).

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Find the intersection point of the linear equations $y = -6x + 2$ and $y = 3x - 7$ .

Assumptions
1. We have two linear equations in the form of $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.
2. We want to find the intersection point of the two lines represented by these equations.

To graph the first equation $y = -6x + 2$ , we need to identify two points that lie on this line. We can do this by choosing two different values for $x$ and calculating the corresponding $y$ values.

Let's choose $x = 0$ and $x = 1$ for the first equation.
For $x = 0$ : $y = -6(0) + 2 = 2$
For $x = 1$ : $y = -6(1) + 2 = -6 + 2 = -4$

Next, we graph the second equation $y = 3x - 7$ in the same way.

Let's choose $x = 0$ and $x = 1$ for the second equation.
For $x = 0$ : $y = 3(0) - 7 = -7$
For $x = 1$ : $y = 3(1) - 7 = 3 - 7 = -4$

By observing the graph, we can determine the coordinates of the intersection point. If the graph is not precise enough, we can find the intersection algebraically by setting the two equations equal to each other and solving for $x$ .
$-6x + 2 = 3x - 7$

Combine like terms by moving all terms involving $x$ to one side and constant terms to the other side.
$-6x - 3x = -7 - 2$

Simplify the equation by combining the terms.
$-9x = -9$

Divide both sides by -9 to solve for $x$ .
$x = \frac{-9}{-9} = 1$

Now that we have the value of $x$ , we can substitute it back into either of the original equations to find the corresponding value of $y$ . Let's use the first equation $y = -6x + 2$ .
$y = -6(1) + 2$

Calculate the value of $y$ .
$y = -6 + 2 = -4$

We have found the intersection point by solving the equations algebraically. The intersection point is $(1, -4)$ .
The intersection point of the two lines represented by the equations $y = -6x + 2$ and $y = 3x - 7$ is $(1, -4)$ .