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Chapter 1: Problem 2

Find the point of intersection of the two lines \(3 x+y=3\) and \(x+2 y=1\).

Short Answer

Expert verified
The point of intersection is (1, 0).

Step by step solution

01

Write down the equations

The two given lines are: 1) \(3x + y = 3\) 2) \(x + 2y = 1\)
02

Express one variable in terms of the other

From the second equation \(x + 2y = 1\), express \(x\) in terms of \(y\): \(x = 1 - 2y\)
03

Substitute the expression into the first equation

Substitute \(x = 1 - 2y\) into the first equation \(3x + y = 3\): \(3(1 - 2y) + y = 3\)
04

Solve for \(y\)

Simplify and solve for \(y\): \(3 - 6y + y = 3\) \(-5y = 0\) \(y = 0\)
05

Solve for \(x\)

Substitute \(y = 0\) back into the expression for \(x\): \(x = 1 - 2(0)\) \(x = 1\)
06

State the point of intersection

The point of intersection is \((1, 0)\).

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Key Concepts

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

Point of Intersection
The point of intersection refers to the exact spot where two lines meet on a graph. To find these points, we look for values of x and y that satisfy both equations simultaneously. An intersection point gives a common solution to both equations.
This exercise presents a practical example: finding the intersection of two lines represented by the equations:
1) \(3x + y = 3\)
2) \(x + 2y = 1\).
After some computation, we find that the lines intersect at the point \((1, 0)\). This means that (1, 0) lies on both lines.
Linear Equations
Linear equations describe relationships between two variables that graph as straight lines. These equations are usually in the form \(ax + by = c\), where a, b, and c are constants. The solution to a linear equation is a set of ordered pairs that make the equation true.
In our example, the equations are:
\(3x + y = 3\)
\(x + 2y = 1\).
These are both linear equations, which can be graphed as straight lines. Their intersection represents the solution that works for both equations.
Substitution Method
The substitution method is an algebraic technique for solving a system of equations. It involves isolating one variable in one equation and substituting its expression into another equation.
Follow these steps to use substitution method:
  • Solve one equation for one variable.
  • Substitute the expression into the other equation.
  • Simplify and solve for the remaining variable.
  • Substitute back to find the first variable.

In our example:
From \(x + 2y = 1\), solve for x:
\(x = 1 - 2y\).
Then, substitute into the first equation:
\(3(1 - 2y) + y = 3\)
Simplify and solve for y:
\(-5y = 0\)
\(y = 0\)
Finally, find x:
\(x = 1 - 2(0)\)
\(x = 1\)
This methods shows that the intersection point is \((1, 0)\).

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Most popular questions from this chapter

If a system of equations has more equations than variables, can it have a solution? If so, give an example and if not, tell why not.

Do the three lines, \(x+2 y=1,2 x-y=1,\) and \(4 x+3 y=3\) have a common point of intersection? If so, find the point and if not, tell why they don't have such a common point of intersection.

Row reduce the following matrix to obtain the row-echelon form. Then continue to obtain the reduced row-echelon form. $$ \left[\begin{array}{rrrr} -2 & 3 & -8 & 7 \\ 1 & -2 & 5 & -5 \\ 1 & -3 & 7 & -8 \end{array}\right] $$

Find h such that $$ \left[\begin{array}{ll|l} 2 & h & 4 \\ 3 & 6 & 7 \end{array}\right] $$ is the augmented matrix of an inconsistent system.

Find the rank of the following matrix. $$ \left[\begin{array}{lllr} 3 & 6 & 5 & 12 \\ 1 & 2 & 2 & 5 \\ 1 & 2 & 1 & 2 \end{array}\right] $$
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