Problem 26 - Algebra 2

Question

The graphs of \(y-2 x=1,4 x+y=7,\) and \(2 y-x=-4\) contain the sides of a triangle. Find the coordinates of the vertices of the triangle.

Step-by-Step Solution

Answer

The vertices of the triangle are (1, 3), (2, -1), and (-2, -3).

Step 1: Understanding the Problem

To find the vertices of the triangle, we need to determine the points of intersection of the lines described by the equations: \(y - 2x = 1\), \(4x + y = 7\), and \(2y - x = -4\). These intersection points will be the vertices of the triangle.

Step 2: Solve for the Intersection of the First Two Lines

Substitute \(y\) from the first equation, \(y = 2x + 1\), into the second equation \(4x + y = 7\): \[4x + (2x + 1) = 7\]Simplifying gives:\[6x + 1 = 7\]\[6x = 6\]\[x = 1\]Substitute \(x = 1\) back into \(y = 2x + 1\) to find \(y\):\[y = 2(1) + 1 = 3\]So, the intersection point is \((1, 3)\).

Step 3: Solve for the Intersection of the Second and Third Lines

From the second equation, express \(y\):\[y = 7 - 4x\]Substitute into the third equation \(2y - x = -4\):\[2(7 - 4x) - x = -4\]Simplify:\[14 - 8x - x = -4\]\[14 - 9x = -4\]\[-9x = -18\]\[x = 2\]Substitute \(x = 2\) back into \(y = 7 - 4x\):\[y = 7 - 4(2) = -1\]The intersection point is \((2, -1)\).

Step 4: Solve for the Intersection of the First and Third Lines

From the first equation, express \(y\):\[y = 2x + 1\]Substitute into the third equation \(2y - x = -4\):\[2(2x + 1) - x = -4\]Simplify:\[4x + 2 - x = -4\]\[3x + 2 = -4\]\[3x = -6\]\[x = -2\]Substitute \(x = -2\) back into \(y = 2x + 1\):\[y = 2(-2) + 1 = -3\]The intersection point is \((-2, -3)\).

Step 5: Conclusion

The vertices of the triangle formed by the given lines are \((1, 3)\), \((2, -1)\), and \((-2, -3)\).

Key Concepts

Systems of Equations

When you have more than one equation that needs to be solved together, you are dealing with a system of equations. In this case, our equations are linear, meaning they represent straight lines on a graph. The main idea is to find values for the variables (in this case, x and y) that satisfy all the equations at the same time. This helps us find points where the lines meet, known as intersections. For example, if we take the equations:

• \(y - 2x = 1\)
• \(4x + y = 7\)
• \(2y - x = -4\)

We solve them systematically to find their common solutions, which represent the triangle's vertices. Systems can be solved using substitution, elimination, or graphing methods.

Line Intersection

The point where two lines cross is called the intersection. Each intersection represents a solution to a system of equations, specifically for the pair of equations that define those lines. In coordinate geometry, this point will have coordinates \(x, y\) that satisfy both equations.Finding the intersection involves solving two equations at a time, either by:

• Substritution: Solving one of the equations for one variable, then inserting that expression into the other equation.
• Elimination: Adding or subtracting the equations to eliminate one variable, making it easier to solve for the other.

For example, to find the intersection of \(y = 2x + 1\) and \(4x + y = 7\), you'd substitute \(y = 2x + 1\) into the second equation and solve for \(x\) and \(y\).

Coordinate Geometry

Coordinate geometry, or analytic geometry, is the study of geometric figures using a coordinate system. It provides tools to describe geometric shapes with numerical data, making it easier to solve problems with algebraic calculations.In this context, the Cartesian coordinate system is particularly useful. It uses two axes (x and y) to specify any point on a plane with an ordered pair \(x, y\). Lines, intersections, and other shapes can all be analyzed using coordinates. For our exercise:

• Each line equation translates to a straight line on this plane, with its own slope and intercept.
• The intersections of these lines determine the vertices of the triangle.

Algebraic Calculation

Algebraic calculation involves manipulating expressions to find unknown values. It is a fundamental part of solving systems of equations and finding line intersections.By substituting values or simplifying equations, we can isolate variables (like x and y) to find precise points of interest, like intersections:

• Substitution: Replace one variable in an equation with an equivalent expression derived from another equation.
• Isolation: Rearrange the equation to get a variable by itself, simplifying as needed.
• Simplification: Perform arithmetic operations to collapse or combine terms for easier solving.

For example, by simplifying the intersection solving process step-by-step, we found the triangle vertices: \(1, 3\), \(2, -1\), and \(-2, -3\). These points are the result of careful algebraic calculation.