Problem 59 - Calculus Single Variable

Question

Find the point at which the lines determined by the two given equations intersect. $$ 2 x+y=4, x+3 y=7 $$

Step-by-Step Solution

Answer

The intersection point is (1, 2).

Step 1: Write the Equations in Slope-Intercept Form

To solve for the intersection, first express each line in slope-intercept form (i.e., solve for \(y\)).\ For the first equation, \(2x + y = 4\), solve for \(y\):\ \[ y = 4 - 2x \] For the second equation, \(x + 3y = 7\), solve for \(y\):\ \[ y = rac{7 - x}{3} \].

Step 2: Set the Equations Equal to Each Other

To find the intersection, set the expressions for \(y\) from both equations equal to each other:\[ 4 - 2x = \frac{7 - x}{3} \].

Step 3: Eliminate the Fraction

Multiply every term in the equation by 3 to eliminate the fraction:\ \[ 3(4 - 2x) = 7 - x \] which simplifies to \[ 12 - 6x = 7 - x \].

Step 4: Solve for x

Rearrange the equation from Step 3 to solve for \(x\):\ Add \(6x\) to both sides to get \(12 = 7 + 5x\). Subtract 7 from both sides:\ \[ 5 = 5x \].Divide by 5 to solve for \(x\):\ \[ x = 1 \].

Step 5: Substitute x Back to Find y

Substitute \(x = 1\) back into either of the original equations to find \(y\). Using \(2x + y = 4\):\ \[ 2(1) + y = 4 \] \[ 2 + y = 4 \] Subtract 2 from both sides to find \(y\): \[ y = 2 \].

Step 6: Verify the Solution

Use the second equation, \(x+3y=7\), to verify our values. Substitute \(x = 1\) and \(y = 2\) into the equation, you get: \ \[ 1 + 3(2) = 7 \] which simplifies to \[ 1 + 6 = 7 \], confirming the solution is correct.

Key Concepts

Systems of Equations

When dealing with two or more equations with multiple variables, you are working with a system of equations. In this scenario, the system consists of two linear equations: \(2x + y = 4\) and \(x + 3y = 7\). Each equation represents a line in a two-dimensional plane. An important task is to determine where, or if, these lines intersect. This intersection point will be the solution that satisfies both equations simultaneously. The goal is to find the values of \(x\) and \(y\) that work for both lines. Systems of equations provide crucial insights into relationships between variables. In real-life applications, they help model and solve practical problems like resource allocation and economic forecasts.

Slope-Intercept Form

The slope-intercept form of a linear equation is a way of writing an equation so that you can easily identify the line's slope and the y-intercept. It is typically written as \(y = mx + b\), where \(m\) represents the slope, and \(b\) stands for the y-intercept. This form helps us quickly understand how steep the line is and where it crosses the y-axis.
To convert the given equations into slope-intercept form, you solve each equation for \(y\). For example, from \(2x + y = 4\), rearrange it to \(y = 4 - 2x\). Similarly, convert \(x + 3y = 7\) to \(y = \frac{7 - x}{3}\). This conversion allows us to easily set the two equations equal to each other to find the intersection point.

Solving Equations

Solving the equations involves finding values for the variables that make both equations true at the same time. Using the slope-intercept forms, both equations for \(y\) can be set equal to each other: \(4 - 2x = \frac{7 - x}{3}\).
To simplify, multiply each term by 3 to clear the fraction, resulting in an equation without fractions: \(12 - 6x = 7 - x\). Then, rearrange and simplify to solve for one variable, \(x\), resulting in \(x = 1\). With the value for \(x\) known, substitute back into one of the original equations to find \(y\), resulting in \(y = 2\). Solving in this manner ensures that the solution found satisfies both original equations.

Verification of Solution

Once a solution is found, verification is a critical step to ensure accuracy. In our case, we determined \(x = 1\) and \(y = 2\). Verification involves substituting these values back into the original equations to see if they hold true.
For \(2x + y = 4\), substitute to get \(2(1) + 2 = 4\), which checks out perfectly. Similarly, when verified against \(x + 3y = 7\), substitute \(1\) and \(2\) to find \(1 + 3(2) = 7\), again confirming the solution. This step ensures no arithmetic errors were made during solving, and the result accurately represents the intersection of the two lines.