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Chapter 4: Problem 92

Construct a system of two linear equations that has a solution of \((-2,6)\)

Short Answer

Expert verified
The system is: 1. \(y = x + 8\) 2. \(y = -2x + 2\)

Step by step solution

01

Understanding the Solution Set

The solution set (-2, 6) means that the point (-2, 6) is where two lines intersect, satisfying both equations in the system. This means when we plug in x = -2 and y = 6 into each equation, they both hold true.
02

Choosing a Line Form

Let's use the slope-intercept form of a line, which is y = mx + c, where m is the slope and c is the y-intercept. We'll construct two different lines that intersect at x = -2 and y = 6.
03

Constructing the First Equation

Choose the slope m_1 = 1. Using the point-slope form y - y_1 = m(x - x_1), where (x_1, y_1) = (-2, 6), we write y - 6 = 1(x + 2). Simplifying, we get y = x + 8.
04

Constructing the Second Equation

Choose a different slope, m_2 = -2. Using the point-slope form again, write y - 6 = -2(x + 2). Simplifying this equation, we obtain y = -2x + 2.
05

Verifying the Solution

Substitute x = -2 into both equations y = x + 8 and y = -2x + 2. For the first equation, y = -2 + 8 = 6, and for the second equation, y = -2(-2) + 2 = 6. Both check out, meaning both lines intersect at (-2, 6).

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Key Concepts

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

Solution of a System
When you have two or more equations, and you are trying to find a common solution, you are solving a system of equations. A solution to a system of linear equations is a point where all the equations are satisfied simultaneously. This means it is the point of intersection of the lines represented by these equations.
This point of intersection is the \((-2, 6)\) in our exercise. Therefore, when you plug in \(-2\) for \ x \ and \ 6 \ for \ y \ into each equation of the system, both equations will become true. So, the key takeaway is:
  • The solution is the point where all lines intersect.
  • Both \ x \ and \ y \ values must satisfy all equations.

Finding this solution involves constructing and verifying equations to ensure they both intersect at this point.
Slope-Intercept Form
The slope-intercept form of a linear equation is \(y = mx + c\). Here 'm' is the slope of the line, and 'c' is the y-intercept. This form is very helpful in quickly identifying how a line behaves and where it crosses the y-axis.
Consider the line \(y = x + 8\):
  • The slope: \(m = 1\) shows that for every step we move along the x-axis, we move an equal step up on the y-axis.
  • The y-intercept: \(c = 8\) tells us that the line crosses the y-axis at \ y = 8 \.

This form is simple and efficient for graphing and understanding the direction and position of the line. In our example, both lines are written in slope-intercept form, making it easy to grasp their intersections and verify solutions.
Point-Slope Form
The point-slope form of a linear equation is given by \(y - y_1 = m(x - x_1)\). This is useful when you know a point on the line \((x_1, y_1)\) and the slope 'm'. You can construct the equation of the line using this information.
For example, if we take \(x_1 = -2\) and \(y_1 = 6\) with slope \(m_1 = 1\), we can form:
\[ y - 6 = 1(x + 2) \] Simplifying this, it transforms into the slope-intercept form and becomes \(y = x + 8\).
  • This form is great for generating equations quickly when a specific point and slope are known.
  • It allows for easy conversion to other forms, like the slope-intercept form, which can be useful for graphing.

The point-slope form is versatile and a handy tool in constructing mathematical representations of linear scenarios.

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Most popular questions from this chapter

Explain the error. Solve: \(\quad\left\\{\begin{array}{l}{3 a+4 b=1} \\ {a+2 b=9}\end{array}\right.\) \(a=9-2 b \quad\) Solve for \(a\) in the second equation. \(9-2 b+2 b=9 \quad\) Substitute \(9-2 b\) for \(a\) \(\begin{aligned} 9=9 & \text { The b-terms drop out. } \\ \text { True } \end{aligned}\) The system has infinitely many solutions.

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {x+y=4} \\ {y=x} \end{array}\right. $$

Cleaning Floors. A custodian is going to mix a \(4 \%\) ammonia solution and a \(12 \%\) ammonia solution to get 1 gallon \((128\) fluid ounces) of a \(9 \%\) ammonia solution. How many fluid ounces of the \(4 \%\) solution and the \(12 \%\) solution should be used?

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {3 x+2 y=-8} \\ {2 x-3 y=-1} \end{array}\right. $$

Solve each system of equations by graphing. $$ \left\\{\begin{array}{l} {y=\frac{3}{4} x+3} \\ {y=-\frac{x}{4}-1} \end{array}\right. $$
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