Question

Solve each system of equations by addition-subtraction, or by substitution. Check some by graphing. $$\begin{aligned} &4 x+2 y=3\\\ &-4 x+y=6 \end{aligned}$$

Short answer

The solution to the system of equations is (-3/4, 3). This can be checked by substituting into both equations and by graphing.

Step-by-step solution

Solve the second equation for y

Take the second equation (-4x + y = 6) and solve for y. You can do this by adding 4x to both sides of the equation, resulting in: y = 4x + 6.

Substitute the expression for y into the first equation

Now substitute y = 4x + 6 into the first equation (4x + 2y = 3). Replace y with 4x + 6 in the first equation to obtain a single equation in terms of x: 4x + 2(4x + 6) = 3.

Solve for x

Expand the equation from Step 2 and solve for x. 4x + 8x + 12 = 3 12x + 12 = 3 Subtract 12 from both sides: 12x = -9 Divide by 12 to find x: x = -9/12 Reduce the fraction: x = -3/4.

Solve for y using the value of x

Substitute x = -3/4 into the equation y = 4x + 6 to solve for y: y = 4(-3/4) + 6 Simplify the equation: y = -3 + 6 Calculate the value of y: y = 3.

Write the solution for the system of equations

The solution is the pair (x, y) that satisfies both equations, which we have found as x = -3/4 and y = 3. So the system of equations solution is (-3/4, 3).

Check the solution by substituting into both original equations

Substitute x = -3/4 and y = 3 into both original equations to verify the solution. For the first equation: 4(-3/4) + 2(3) = 3 checks out. For the second equation: -4(-3/4) + 3 = 6 also checks out. Therefore, the solution is correct.

Check the solution by graphing

Graph both equations on the same coordinate plane. The point where the lines intersect should be the solution of the system. If the intersection point is (-3/4, 3), then the graphical method confirms our algebraic solution.

Key concepts

Addition-Subtraction Method

The addition-subtraction method is a foundational technique to solve systems of linear equations, where one equation is manipulated and then added to or subtracted from the other to eliminate one of the variables. To use this method effectively:

• Identify the coefficients of the variables in both equations.
• Manipulate the equations by multiplying them with suitable numbers to get equal coefficients for one variable with opposite signs.
• Add or subtract the equations to eliminate that variable, thus reducing the system to a single equation with one variable.
• Solve for the remaining variable and back-substitute to find the value of the eliminated variable.

For example, the equations \(4x + 2y = 3\) and \(-4x + y = 6\) are well-suited for this method since \(4x\) and \(-4x\) are already opposites — no manipulation needed here. By adding the equations, the \(x\) variable is eliminated and you can easily solve for \(y\), then back-substitute to find \(x\).
Remember, the key is to make the coefficients of one of the variables opposites, which cancels that variable when adding or subtracting the equations.

Substitution Method

The substitution method is another popular approach for solving systems of equations, especially if one of the equations can easily be solved for one variable. Here's how to tackle it step by step:

• Isolate one variable in one of the equations.
• Substitute this expression into the other equation.
• Solve the resulting single-variable equation.
• Then, use the found value to solve for the initial isolated variable.

Considering our textbook system, isolating \(y\) in the second equation (as done in Step 1) and substituting it into the first (as in Step 2) streamlines the process. This reduces the system down to a single variable equation, which can be quickly solved. The substitution method works especially well when one equation is already solved for a variable or can be solved easily, as was the case here. It turns a system of equations into successive single-variable problems, simplifying the algebra involved.

Graphical Method of Solving Systems

The graphical method of solving systems involves drawing the equations as lines on a coordinate plane and finding the intersection point. This visual approach can be used to confirm solutions found algebraically. The steps include:

• Rewrite each equation in slope-intercept form (\(y = mx + b\)), if possible.
• Graph each line on the same coordinate plane, using the slope and y-intercept.
• Identify the point of intersection, which represents the solution to the system.

In our case, the first equation would convert to \(y = -2x + \frac{3}{2}\) and the second to \(y = 4x + 6\). These lines can be graphed and where they cross confirms our solution (\(x = -3/4\), \(y = 3\)). The graphical method provides a good cross-reference against algebraic solutions, though it is not as precise because it relies on the accuracy of the drawing and can be difficult if the intersection isn't at an exact coordinate point.

Algebraic Manipulation

Algebraic manipulation is a broad skill set used in solving equations, encompassing a variety of techniques like adding, subtracting, multiplying, dividing, and factoring. To use these tools effectively when solving systems:

• Understand the properties of equality and operations to maintain balanced equations.
• Perform operations step by step, simplifying equations to make variables more manageable.
• Keep your work organized to avoid errors in the manipulation process.

In the solution to our system, Steps 3 and 4 demonstrate algebraic manipulation. After substituting \(y\)'s expression, the first equation was expanded and simplified to solve for \(x\). When \(x\)'s value was found, it was substituted back to find \(y\), showcasing a clean execution of algebraic manipulation techniques. Mastery of these techniques is crucial for solving all types of algebraic problems, including system of equations.