Math Quiz: Systems Of Equations!

The correct answer is "one solution at (1, 2)". To determine the solution, we can graph the system of equations. The first equation, 3x - 2y = -1, can be rearranged to y = (3/2)x + (1/2), which is a linear equation with a slope of 3/2 and a y-intercept of 1/2. The second equation, 2x + y = 4, can be rearranged to y = -2x + 4, which is a linear equation with a slope of -2 and a y-intercept of 4. By graphing these two equations, we find that they intersect at the point (1, 2), indicating that there is one solution.

The correct answer is (6, -3). To solve the system by substitution, we can solve one equation for one variable and substitute it into the other equation. From the second equation, we can solve for x in terms of y: x = y + 9. Substituting this value into the first equation, we get 2(y + 9) + 3y = 3. Simplifying, we have 2y + 18 + 3y = 3, which gives us 5y + 18 = 3. Solving for y, we find y = -3. Substituting this value back into the equation x = y + 9, we get x = -3 + 9, which gives us x = 6. Therefore, the solution to the system is (6, -3).

To find the value of x that satisfies the equation 2x^2 + 4x + 2 = 0, we can use the quadratic formula. Plugging in the coefficients a = 2, b = 4, and c = 2 into the formula, we get x = (-4 ± √(4^2 - 4(2)(2))) / (2(2)). Simplifying this expression gives us x = (-4 ± √(16 - 16)) / 4, which further simplifies to x = (-4 ± √0) / 4. Since the square root of 0 is 0, we have x = (-4 ± 0) / 4, which simplifies to x = -4 / 4 = -1. Therefore, -1 is the correct answer.

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When solving by elimination, the first step is to distribute. This means multiplying each term in one equation by a constant so that the coefficients of one of the variables in both equations become opposites. This allows us to eliminate one variable when we add or subtract the equations.

When you solve a system of equations, there can be three possible types of solutions. The first type is a unique solution, where the two equations intersect at a single point. The second type is no solution, where the two equations are parallel and do not intersect. The third type is infinitely many solutions, where the two equations are the same and overlap each other. Therefore, there are three possible types of solutions when solving a system of equations.

The equation of a line can be written in the form y = mx + b, where m is the gradient of the line and b is the y-intercept. Given that the gradient is -2/3 and the line passes through the point (3,0), we can substitute these values into the equation to find the y-intercept. Plugging in x = 3 and y = 0, we get 0 = (-2/3)(3) + b. Solving for b, we find that b = 2. Therefore, the equation of the line is y = -2/3x + 2.

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